Knot contact homology, string topology, and the cord algebra · Tome 4, 2017, p.661–780 DOI:...

Kai Cieliebak, Tobias Ekholm, Janko Latschev, & Lenhard NgKnot contact homology, string topology, and the cord algebraTome 4 (2017), p. 661-780.

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Tome 4, 2017, p. 661–780 DOI: 10.5802/jep.55

KNOT CONTACT HOMOLOGY, STRING TOPOLOGY,

AND THE CORD ALGEBRA

by Kai Cieliebak, Tobias Ekholm, Janko Latschev& Lenhard Ng

Abstract. — The conormal Lagrangian LK of a knot K in R3 is the submanifold of thecotangent bundle T ∗R3 consisting of covectors along K that annihilate tangent vectors to K.By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK ,and the Legendrian contact homology of ΛK is a knot invariant of K, known as knot contacthomology. We define a version of string topology for strings in R3 ∪ LK and prove that thisis isomorphic in degree 0 to knot contact homology. The string topology perspective gives atopological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology)and relates it to the knot group. Together with the isomorphism this gives a new proof thatknot contact homology detects the unknot. Our techniques involve a detailed analysis of certainmoduli spaces of holomorphic disks in T ∗R3 with boundary on R3 ∪ LK .

Résumé (Homologie de contact pour les nœuds, topologie des cordes et algèbre des cordes)Le fibré conormal lagrangien LK d’un nœud K dans R3 est la sous-variété du fibré cotangent

T ∗R3 formée des covecteurs le long de K qui annulent les vecteurs tangents à K. En l’inter-sectant avec le fibré cotangent unitaire S∗R3, on obtient le fibré conormal unitaire ΛK , dontl’homologie de contact legendrienne est un invariant du nœud K, appelé homologie de contactpour les nœuds. Nous définissons une version de la topologie des cordes pour des cordes dansR3∪LK et montrons qu’elle est isomorphe en degré 0 à l’homologie de contact pour les nœuds.La topologie des cordes permet une approche topologique de l’algèbre des cordes (qui est aussiisomorphe à l’homologie de contact pour les nœuds en degré 0) et la relie au groupe du nœud.Ceci donne, joint à cet isomorphisme, une nouvelle démonstration du fait que l’homologie decontact pour les nœuds détecte le nœud trivial. Nos techniques font intervenir une analysedétaillée de certains espaces de modules de disques holomorphes dans T ∗R3 avec bord dansR3 ∪ LK .

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622. String homology in degree zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6653. Roadmap to the proof of Theorem 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6814. Holomorphic functions near corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

Mathematical subject classification (2010). — 53D42, 55P50, 57R17, 57M27.Keywords. — Holomorphic curve, string topology, conormal bundle, knot invariant, Lagrangian sub-manifold, Legendrian submanifold.

The work of KC was supported by DFG grants CI 45/2-1 and CI 45/5-1. The work of TE wassupported by the Knut and Alice Wallenberg Foundation and by the Swedish Research Council. Thework of JL was supported by DFG grant LA 2448/2-1. The work of LN was supported by NSF grantDMS-1406371 and a grant from the Simons Foundation (# 341289 to Lenhard Ng).

e-ISSN: 2270-518X http://jep.cedram.org/

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662 K. Cieliebak, T. Ekholm, J. Ekholm & L. Ng

5. String homology in arbitrary degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6876. The chain map from Legendrian contact homology to string homology . . . . . . . 6997. Proof of the isomorphism in degree zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7198. Properties of holomorphic disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7389. Transversely cut out solutions and orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75010. Compactification of moduli spaces and gluing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

1. Introduction

To a smooth n-manifold Q we can naturally associate a symplectic manifold anda contact manifold: its cotangent bundle T ∗Q with the canonical symplectic struc-ture ω = dp ∧ dq, and its unit cotangent bundle (with respect to any Riemannianmetric) S∗Q ⊂ T ∗Q with its canonical contact structure ξ = ker(p dq). Moreover, ak-dimensional submanifold K ⊂ Q naturally gives rise to a Lagrangian and a Legen-drian submanifold in T ∗Q resp. S∗Q: its conormal bundle

LK = (q, p) ∈ T ∗Q | q ∈ K, p|TqK = 0

and its unit conormal bundle ΛK = LK∩S∗Q. Symplectic field theory (SFT [20]) pro-vides a general framework for associating algebraic invariants to a pair (M,Λ) consist-ing of a contact manifold and a Legendrian submanifold; when applied to (S∗Q,ΛK),these invariants will be diffeotopy invariants of the manifold pair (Q,K). The studyof the resulting invariants was first suggested by Y. Eliashberg.

In this paper we concentrate on the case where K is a framed oriented knot inQ = R3. Moreover, we consider only the simplest SFT invariant: Legendrian contacthomology. For Q = R3, S∗Q is contactomorphic to the 1-jet space J1(S2), for whichLegendrian contact homology has been rigorously defined in [15]. The Legendriancontact homology of the pair (S∗R3,ΛK) is called the knot contact homology of K.We will denote it Hcontact

∗ (K).In its most general form (see [11, 33]), knot contact homology is the homology of a

differential graded algebra over the group ring Z[H2(S∗R3,ΛK)] = Z[λ±1, µ±1, U±1],where the images of λ, µ under the connecting homomorphism generate H1(ΛK) =

H1(T 2) and U generates H2(S∗R3). The isomorphism class of Hcontact∗ (K) as a

Z[λ±1, µ±1, U±1]-algebra is then an isotopy invariant of the framed oriented knot K.The topological content of knot contact homology has been much studied in recent

years; see for instance [1] for a conjectured relation, which we will not discuss here, tocolored HOMFLY-PT polynomials and topological strings. One part of knot contacthomology that has an established topological interpretation is its U = 1 specialization.In [31, 32], the third author constructed a knot invariant called the cord algebraCord(K), whose definition we will review in Section 2.2. The combined results of[31, 32, 12] then prove that the cord algebra is isomorphic as a Z[λ±1, µ±1]-algebra

J.É.P. — M., 2017, tome 4

Knot contact homology, string topology, and the cord algebra 663

to the U = 1 specialization of degree 0 knot contact homology. We will assumethroughout this paper that we have set U = 1;(1) then the result is:

Theorem 1.1 ([31, 32, 12]). — Hcontact0 (K) ∼= Cord(K).

It has been noticed by many people that the definition of the cord algebra bearsa striking resemblance to certain operations in string topology [4, 36]. Indeed, Basu,McGibbon, Sullivan, and Sullivan used this observation in [2] to construct a theorycalled “transverse string topology” associated to any codimension 2 knot K ⊂ Q, andproved that it determines the U = λ = 1 specialization of the cord algebra.

In this paper, we present a different approach to knot contact homology and thecord algebra via string topology. Motivated by the general picture sketched by thefirst and third authors in [6], we use string topology operations to define the stringhomology Hstring

∗ (K) of K. Then the main result of this paper is:

Theorem 1.2. — For any framed oriented knot K ⊂ R3, we have an isomorphismbetween U = 1 knot contact homology and string homology in degree 0,

Hcontact0 (K) ∼= Hstring

0 (K),

defined by a count of punctured holomorphic disks in T ∗R3 with Lagrangian boundarycondition LK ∪ R3.

On the other hand, degree 0 string homology is easily related to the cord algebra:

Proposition 1.3. — For any framed oriented knot K ⊂ R3, we have an isomorphism

Hstring0 (K) ∼= Cord(K).

As a corollary we obtain a new geometric proof of Theorem 1.1. In fact, we evenprove a slight refinement of the usual formulation of Theorem 1.1, as we relate certainnoncommutative versions of the two sides where the coefficients λ, µ do not commutewith everything; see Section 2.2 for the version of Cord(K) and Section 6.2 for thedefinition of Hcontact

0 (K) that we use.Our proof is considerably more direct than the original proof of Theorem 1.1,

which was rather circuitous and went as follows. The third author constructed in[30, 32] a combinatorial differential graded algebra associated to a braid whose closureis K, and then proved in [31, 32] that the degree 0 homology of this combinatorialcomplex is isomorphic to Cord(K) via a mapping class group argument. The secondand third authors, in joint work with Etnyre and Sullivan [12], then proved thatthe combinatorial complex is equal to the differential graded algebra for knot contacthomology, using an analysis of degenerations of holomorphic disks to Morse flow trees.

Besides providing a cleaner proof of Theorem 1.1, the string topology formulationalso gives a geometric explanation for the somewhat mystifying skein relations that

(1)However, we note that it is an interesting open problem to find a similar topological interpre-tation of the full degree 0 knot contact homology as a Z[λ±1, µ±1, U±1]-algebra.

J.É.P. — M., 2017, tome 4


define the cord algebra. Moreover, string homology can be directly related to thegroup ring Zπ of the fundamental group π = π1(R3 rK) of the knot complement:

Proposition 1.4 (see Proposition 2.21). — For a framed oriented knot K ⊂ R3,Hstring

0 (K) ∼= Hcontact0 (K) is isomorphic to the subring of Zπ generated by λ±1, µ±1,

and im(1−µ), where λ, µ are the elements of π representing the longitude and merid-ian of K, and 1− µ denotes the map Zπ → Zπ given by left multiplication by 1− µ.

As an easy consequence of Proposition 1.4, we recover the following result from [32]:

Corollary 1.5 (see Section 2.4). — Knot contact homology detects the unknot: ifHcontact

0 (K) ∼= Hcontact0 (U) where K is a framed oriented knot in R3 and U is the

unknot with any framing, then K = U as framed oriented knots.

The original proof of Corollary 1.5 in [32] uses the result that the A-polynomial detectsthe unknot [8], which in turn relies on results from gauge theory [28]. By contrast, ourproof of Corollary 1.5 uses no technology beyond the Loop Theorem (more precisely,the consequence of the Loop Theorem that the longitude is null-homotopic in R3 rKif and only if K is unknotted).

Organization of the paper. — In Section 2 we define degree 0 string homology andprove Proposition 1.3, Proposition 1.4 and Corollary 1.5. The remainder of the paperis occupied by the proof of Theorem 1.2, beginning with an outline in Section 3. Aftera digression in Section 4 on the local behavior of holomorphic functions near corners,which serves as a model for the behavior of broken strings at switches, we define stringhomology in arbitrary degrees in Section 5.

The main work in proving Theorem 1.2 is an explicit description of the modulispaces of holomorphic disks in T ∗R3 with boundary on LK ∪ R3 and punctures as-ymptotic to Reeb chords. In Section 6 we state the main results about these modulispaces and show how they give rise to a chain map from Legendrian contact homologyto string homology (in arbitrary degrees). Moreover, we show that this chain map re-spects a natural length filtration. In Section 7 we construct a length decreasing chainhomotopy and prove Theorem 1.2.

The technical results about moduli spaces of holomorphic disks and their com-pactifications as manifolds with corners are proved in the remaining Sections 8, 9and 10.

Extensions. — The constructions in this paper have several possible extensions.Firstly, the definition of string homology and the construction of a homomorphismfrom Legendrian contact homology to string homology in degree zero work the sameway for a knot K in an arbitrary 3-manifold Q instead of R3 (the correspondingsections are actually written in this more general setting), and more generally fora codimension 2 submanifold K of an arbitrary manifold Q.(2) The fact that the

(2)In the presence of contractible closed geodesics in Q, this will require augmentations by holo-morphic planes in T ∗Q, see e.g. [6].

J.É.P. — M., 2017, tome 4


ambient manifold is R3 is only used to obtain a certain finiteness result in the proofthat this map is an isomorphism (see Remark 7.9). If this result can be generalized,then Theorem 1.2 will hold for arbitrary codimension 2 submanifolds K ⊂ Q.

Secondly, for knots in 3-manifolds, the homomorphism from Legendrian contacthomology to string homology is actually constructed in arbitrary degrees. Provingthat it is an isomorphism in arbitrary degrees will require analyzing codimensionthree phenomena in the space of strings with ends on the knot, in addition to thecodimension one and two phenomena described in this paper.

Acknowledgments. — We thank Chris Cornwell, Tye Lidman, and especially YashaEliashberg for stimulating conversations. This project started when the authors metat the Workshop “SFT 2” in Leipzig in August 2006, and the final technical detailswere cleaned up when we met during the special program on “Symplectic geometryand topology” at the Mittag-Leffler institute in Djursholm in the fall of 2015. Wewould like to thank the sponsors of these programs for the opportunities to meet, aswell as for the inspiring working conditions during these events. Finally, we thank thereferee for suggesting numerous improvements.

2. String homology in degree zero

In this section, we introduce the degree 0 string homology Hstring0 (K). The dis-

cussion of string homology here is only a first approximation to the more preciseapproach in Section 5, but is much less technical and suffices for the comparison tothe cord algebra. We then give several formulations of the cord algebra Cord(K) anduse these to prove that Hstring

0 (K) ∼= Cord(K) and that string homology detects theunknot. Throughout this section, K denotes an oriented framed knot in some oriented3-manifold Q.

2.1. A string topology construction. — Here we define Hstring0 (K) for an oriented

knot K ⊂ Q. Let N be a tubular neighborhood of K. For this definition we do notneed a framing for the knot K; later, when we identify Hstring

0 (K) with the cordalgebra, it will be convenient to fix a framing, which will in turn fix an identificationof N with S1 ×D2.

Any tangent vector v to Q at a point on K has a tangential component parallelto K and a normal component lying in the disk fiber; write vnormal for the normalcomponent of v. Fix a base point x0 ∈ ∂N and a unit tangent vector v0 ∈ Tx0∂N .

Definition 2.1. — A broken (closed) string with 2` switches on K is a tuple s =

(a1, . . . , a2`+1; s1, . . . , s2`+1) consisting of real numbers 0 = a0 < a1 < · · · < a2`+1

and C1 maps

s2i+1 : [a2i, a2i+1] −→ N, s2i : [a2i−1, a2i] −→ Q

satisfying the following conditions:(i) s0(0) = s2`+1(a2`+1) = x0 and s0(0) = s2`+1(a2`+1) = v0;(ii) for j = 1, . . . , 2`, sj(aj) = sj+1(aj) ∈ K;

J.É.P. — M., 2017, tome 4


x0v0s1

s2

s3

s4

s5K

Figure 2.1. A broken closed string with 4 switches. Here, as in subse-quent figures, we draw the knot K in black, Q-strings (s2, s4) in red,and N -strings (s1, s3, s5) in blue (dashed for clarity to distinguishfrom the red Q-strings).

(iii) for i = 1, . . . , `,

(s2i(a2i))normal = −(s2i+1(a2i))

normal

(s2i−1(a2i−1))normal = (s2i(a2i−1))normal.

We will refer to the s2i and s2i+1 as Q-strings and N-strings, respectively. Denoteby Σ` the set of broken strings with 2` switches.

The last condition, involving normal components of the tangent vectors to the endsof the Q- and N -strings, models the boundary behavior of holomorphic disks in thiscontext (see Sections 4.1 and 5.1 for more on this point). A typical picture of a brokenstring is shown in Figure 2.1.

We call a broken string s = (s1, . . . , s2`+1) generic if none of the derivativessi(ai−1), si(ai) is tangent to K and no si intersects K away from its end points.We call a smooth 1-parameter family of broken strings sλ = (sλ1 , . . . , s

λ2`+1), λ ∈ [0, 1],

generic if s0 and s1 are generic strings, none of the derivatives sλi (aλi−1), sλi (aλi ) istangent to K, and for each i the family sλi intersects K transversally in the interior.The boundary of this family is given by

∂sλ := s1 − s0.

We define string coproducts δQ and δN as follows, cf. Section 5.3. Fix a familyof bump functions (which we will call spikes) sν : [0, 1] → D2 for ν ∈ D2 suchthat s−1

ν (0) = 0, 1, sν(0) = ν and sν(1) = −ν; for each ν, sν lies in the linejoining 0 to ν. For a generic 1-parameter family of broken strings sλ denote byλj , bj the finitely many values for which sλ

j

2i (bj) ∈ K for some i = i(j). For each j,let sj = sνj (· − bj) : [bj , bj + 1]→ N be a shift of the spike associated to the normalderivative νj := −(σλ

j

2i (bj))normal, with constant value sλj2i (bj) along K; interpret this

as an N -string in the normal disk to K at the point sλj2i (bj), traveling along the line

J.É.P. — M., 2017, tome 4


λ = 1λ = 0

sλ

1

2

3

K K

δN

λ = 1λ = 0

sλ

1

2

3

K K

δQ

Figure 2.2. The definition of δN and δQ. The two configurationsshown have sign ε = 1. If the orientation of the 1-parameter fam-ily sλ is switched, i.e., the λ = 0 and λ = 1 ends are interchanged,then δN and δQ are still as shown, but with sign ε = −1. The coor-dinate axes denote orientations chosen on N (top) and Q (bottom).

joining 0 to νj ∈ D2. Now set

δQsλ :=∑j

εj(sλ

j

1 , . . . , sλj

2i |[a2i−1,bj ], sj , sλ

j

2i |[bj ,a2i], . . . , sλj

2`+1

),

where the hat means shift by 1 in the argument, and εj = ±1 are signs defined asin Figure 2.2.(3) Loosely speaking, δQ inserts an N -spike at all points where someQ-string meets K, in such a way that (iii) still holds. The operation δN is definedanalogously, inserting a Q-spike where an N -string meets K (and defining νj withoutthe minus sign).

Denote by C0(Σ`) and C1(Σ`) the free Z-modules generated by generic brokenstrings and generic 1-parameter families of broken strings with 2` switches, respec-tively, and set

Ci(Σ) :=∞⊕=0

Ci(Σ`), i = 0, 1.

Concatenation of broken strings at the base point gives C0(Σ) the structure of a(noncommutative but strictly associative) algebra over Z. The operations defined

(3)Regarding the signs: from our considerations of orientation bundles in Section 9, we can assignthe same sign (which we have chosen to be ε = 1) to both configurations shown in Figure 2.2,provided we choose orientations on Q and N appropriately. More precisely, at a point p on K, if(v1, v2, v3) is a positively oriented frame in Q where v1 is tangent to K and v2, v3 are normal to K,then we need (v1, Jv2,−Jv3) to be a positively oriented frame in N , where J is the almost complexstructure that rotates normal directions in Q to normal directions in N . As a result, if we give Q anyorientation and view N as the subset of Q given by a tubular neighborhood of K, then we assignthe opposite orientation to N .

J.É.P. — M., 2017, tome 4


above yield linear maps

∂ : C1(Σ`) −→ C0(Σ`) ⊂ C0(Σ), δN , δQ : C1(Σ`) −→ C0(Σ`+1) ⊂ C0(Σ).

Define the degree zero string homology of K as

Hstring0 (K) = H0(Σ) := C0(Σ)/ im(∂ + δN + δQ).

Since ∂ + δN + δQ commutes with multiplication by elements in C0(Σ), its image isa two-sided ideal in C0(Σ). Hence degree zero string homology inherits the structureof an algebra over Z. By definition, Hstring

0 (K) is an isotopy invariant of the orientedknot K (the framing was used only for convenience but is not really needed for theconstruction, cf. Remark 2.3 below).

Considering 1-parameter families consisting of generic strings (on which δN and δQvanish), we see that for the computation of Hstring

0 (K) we may replace the algebraC0(Σ) by its quotient under homotopy of generic strings. On the other hand, if sλis a generic 1-parameter family of strings that consists of generic strings except for anN -string (resp. a Q-string) that passes through K exactly once, then δN (resp. δQ)contributes a term to (∂ + δN + δQ), and setting (∂ + δN + δQ)(sλ) = 0 in thesetwo cases yields the following “skein relations”:

(a) 0 =

K

−

K

+

K

(b) 0 =

K

−

K

+

K

.

Since any generic 1-parameter family of broken closed strings can be divided into 1-parameter families each of which crossesK at most once, we have proved the followingresult.

Proposition 2.2. — Let B be the quotient of C0(Σ) by homotopy of generic brokenstrings and let J ⊂ B be the two-sided ideal generated by the skein relations (a)and (b). Then

Hstring0 (K) ∼= B/J .

Remark 2.3. — Degree zero string homology Hstring0 (as well as its higher degree

version defined later) is an invariant of an oriented knot K ⊂ Q. Reversing the orien-tation of K has the result of changing the signs of δN and δQ but not of ∂ and givesrise to isomorphic Hstring

0 . More precisely, if −K is K with the opposite orientation,the map C0(Σ) → C0(Σ) given by multiplication by (−1)` on the summand C0(Σ`)

intertwines the differentials ∂ + δN + δQ for K and −K and induces an isomorphismHstring

0 (K) → Hstring0 (−K). Similarly, mirroring does not change Hstring

0 up to iso-morphism: if K is the mirror of K, then the mirror (reflection) map induces a mapC0(Σ) → C0(Σ), and composing with the above map C0(Σ) → C0(Σ) gives a chainisomorphism C0(Σ)→ C0(Σ).

J.É.P. — M., 2017, tome 4


x0

v0s1 s2

s3

s4s5

K

Figure 2.3. In the alternate definition that produces modified stringhomology, a broken closed string with 4 switches. As usual, Q-stringsare in red, N -strings in (dashed) blue.

In Sections 2.2 through 2.4, we will “improve” Hstring0 from an abstract ring to one

that canonically contains the ring Z[λ±1, µ±1]. This requires a choice of framing of K(though for Q = R3, there is a canonical choice given by the Seifert framing). In theimproved setting, Hstring

0 changes under orientation reversal of K by replacing (λ, µ)

by (λ−1, µ−1); under framing change by f ∈ Z by replacing (λ, µ) by (λµf , µ); andunder mirroring by replacing (λ, µ) by (λ, µ−1). In particular, the improved Hstring

0 isvery sensitive to framing change and mirroring. For a related discussion, see [32, §4.1].

A modified version of string homology. — The choice of the base point in N ratherthan Q in the definition of string homology Hstring

0 (K) is dictated by the relation toLegendrian contact homology. However, from the perspective of string topology wecould equally well pick the base point in Q, as we describe next.

Choose a base point x0 ∈ Q r K and a tangent vector v0 ∈ Tx0Q. Modify thedefinition of a broken string with 2` switches to s = (a0, . . . , a2`+1; s0, . . . , s2`), wherenow

s2i : [a2i, a2i+1] −→ Q, s2i−1 : [a2i−1, a2i] −→ N,

and we require that s0(a0) = s2`(a2`+1) = x0, s0(a0) = s2`(a2`+1) = v0 and condi-tions (ii) and (iii) of Definition 2.1 hold. See Figure 2.3.

Let C0(Σ) denote the ring generated as a Z-module by generic broken strings withbase point x0 ∈ Q. (As usual, the product operation on C0(Σ) is given by stringconcatenation.) We can define string coproducts δN , δQ as before, and then definethe degree 0 modified string homology of K as

Hstring0 (K) = C0(Σ)/ im(∂ + δN + δQ).

We have the following analogue of Proposition 2.2.

Proposition 2.4. — Let B be the quotient of C0(Σ) by homotopy of generic brokenstrings and let J ⊂ B be the two-sided ideal generated by the skein relations (a)and (b). Then

Hstring0 (K) ∼= B/J .

There is one key difference between Hstring0 and Hstring

0 . Since any elementin π1(Q r K,x0) can be viewed as a pure Q-string, we have a canonical map

J.É.P. — M., 2017, tome 4


Zπ1(Q rK,x0) → Hstring0 (K). In fact, we will see in Proposition 2.17 that this is a

ring isomorphism. The same is not the case for Hstring0 (K).

2.2. The cord algebra. — The definition of Hstring0 (K) in Section 2.1 is very similar

to the definition of the cord algebra of a knot [31, 32, 34]. Here we review the cordalgebra, or more precisely, present a noncommutative refinement of it, in which the“coefficients” λ, µ do not commute with the “cords”.

Let K ⊂ Q be an oriented knot equipped with a framing, and let K ′ be a par-allel copy of K with respect to this framing. Choose a base point ∗ on K and acorresponding base point ∗ on K ′ (in fact only the base point on K ′ will be needed).

Definition 2.5. — A (framed) cord of K is a continuous map γ : [0, 1] → Q suchthat γ([0, 1]) ∩K = ∅ and γ(0), γ(1) ∈ K ′ r ∗. Two framed cords are homotopic ifthey are homotopic through framed cords.

We now construct a noncommutative unital ring A as follows: as a ring, A is freelygenerated by homotopy classes of cords and four extra generators λ±1, µ±1, modulothe relations

λ · λ−1 = λ−1 · λ = µ · µ−1 = µ−1 · µ = 1, λ · µ = µ · λ.

Thus A is generated as a Z-module by (noncommutative) words in homotopy classesof cords and powers of λ and µ (and the powers of λ and µ commute with each other,but not with any cords).

Definition 2.6. — The cord algebra of K is the quotient ring

Cord(K) = A /I ,

where I is the two-sided ideal of A generated by the following “skein relations”:

(i) = 1− µ

(ii) = µ · and = · µ

(iii)*

* = λ ·*

* and*

* =*

* · λ

(iv) − = · .

Here K is depicted in black and K ′ parallel to K in gray, and cords are drawn in red.

Remark 2.7. — The skein relations in Definition 2.6 depict cords in space that agreeoutside of the drawn region (except in (iv), where either of the two cords on theleft hand side of the equation splits into the two on the right). Thus (ii) states that

J.É.P. — M., 2017, tome 4


appending a meridian to the beginning or end of a cord multiplies that cord by µ onthe left or right, and (iv) is equivalent to:

− = · .

Remark 2.8. — Our stipulation that λ, µ not commute with cords necessitates adifferent normalization of the cord algebra ofK ⊂ Q from previous definitions [32, 34].In the definition from [34] ([32] is the same except for a change of variables), λ, µcommute with cords, and the parallel copy K ′ is not used. Instead, cords are definedto be paths that begin and end on K with no interior point lying on K, and the skeinrelations are suitably adjusted, with the key relation, the equivalent of (iv), being:

− µ = · .

Let Cord′(K) denote the resulting version of cord algebra.If we take the quotient of the cord algebra Cord(K) from Definition 2.6 where

λ, µ commute with everything, then the result is a Z[λ±1, µ±1]-algebra isomorphic toCord′(K), as long as we take the Seifert framing (lk(K,K ′) = 0). The isomorphismis given as follows: given a framed cord γ, extend γ to an oriented closed loop γ inQ rK by joining the endpoints of γ along K ′ in a way that does not pass throughthe base point ∗, and map γ to µ− lk(γ,K)γ. This is a well-defined map on Cord(K)

and sends the relations for Cord(K) to the relations for Cord′(K). See also the proofof Theorem 2.10 in [32].

We now show that the cord algebra is exactly equal to degree 0 string homology.This follows from the observation that the Q-strings in a generic broken closed stringare each a framed cord of K, once we push the endpoints of the Q-string off of K;and thus a broken closed string can be thought of as a product of framed cords.

Proposition 2.9. — Let K ⊂ Q be a framed oriented knot. Then we have a ringisomorphism

Cord(K) ∼= Hstring0 (K).

Proof. — Choose a normal vector field v along K defining the framing and let K ′be the pushoff of K in the direction of v, placed so that K ′ lies on the boundary ofthe tubular neighborhood N of K. Fix a base point p 6= ∗ on K, and let p′ be thecorresponding point on K ′, so that v(p) is mapped to p′ under the diffeomorphismbetween the normal bundle to K and N . Identify p′ with x0 ∈ ∂N from Definition 2.1(the definition of broken closed string). We can homotope any cord of K so that itbegins and ends at p′, by pushing the endpoints of the cord along K ′, away from ∗,until they reach p′.

Every generator of Cord(K) as a Z-module has the form α1x1α2x2 · · ·x`α`+1,where ` > 0, x1, . . . , x` are cords of K, and α1, . . . , α`+1 are each of the form λaµb

for a, b ∈ Z. We can associate a broken closed string with 2` switches as follows.

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K K K K

K′ K′ K′ K′

p p p p

p′ p′ p′ p′xk xk

γQ

γQ

αk αk

γN γN

Figure 2.4. Turning an element of the cord algebra into a brokenclosed string.

Assume that each cord x1, . . . , x` begins and ends at p′. Fix paths γQ, γQ in Q

from p, p′ to p′, p respectively, and paths γN , γN in N from p, p′ to p′, p respectively,as shown in Figure 2.4: these are chosen so that the derivative of γQ, γQ, γN , γN at pis −v(p),−v(p), v(p),−v(p), respectively. For k = 1, . . . , `, let xk be the Q-string withendpoints at p given by the concatenation γQ · xk · γQ (more precisely, smoothen thisstring at p′). Similarly, for k = 1, . . . , ` + 1, identify αk ∈ π1(∂N) = π1(T 2) with aloop in ∂N with basepoint p′ representing this class; then define αk to be the N -stringγN · αk · γN for k = 1, . . . , `, α1 · γN for k = 0, and γN · α`+1 for k = `+ 1. (If ` = 0,then α1 = α1.) Then the concatenation

α1 · x1 · α2 · x2 · · ·x` · α`+1

is a broken closed string with 2` switches.Extend this map from generators of Cord(K) to broken closed strings to a map

on Cord(K) by Z-linearity. We claim that this induces the desired isomorphism φ :

Cord(K) → Hstring0 (K). Recall that Cord(K) is defined by skein relations (i), (ii),

(iii), (iv) from Definition 2.6, while Hstring0 (K) is defined by skein relations (a), (b)

from Proposition 2.2.To check that φ is well-defined, we need for the skein relations (i), (ii), (iii), (iv) to

be preserved by φ. Indeed, (i) maps under φ to

= − ,

which holds in Hstring0 (K) since both sides are equal to : the left hand side by

rotating the end of the red Q-string and the beginning of the blue N -string around Kat their common endpoint, the right hand side by skein relation (a). Skein relation (iv)maps under φ to

− = ,

which holds by (b). Finally, (ii) and (iii) map to homotopies of broken closed strings:for instance, the left hand relation in (ii) maps to

= .

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To show that φ is an isomorphism, we simply describe the inverse map from brokenclosed strings to the cord algebra. Given any broken closed string, homotope it so thatthe switches all lie at p, and so that the tangent vector to the endpoint of all stringsending at p is −v(p); then the result is in the image of φ by construction. There ismore than one way to homotope a broken closed string into this form, but any suchform gives the same element of the cord algebra: moving the switches along K to pin a different way gives the same result by (iii), while moving the tangent vectorsto −v(p) in a different way gives the same result by (ii). The two skein relations (a)and (b) are satisfied in the cord algebra because of (i) and (iv).

As mentioned in the Introduction, when Q = R3, it is an immediate consequenceof Theorem 1.2 and Proposition 2.9 that the cord algebra is isomorphic to degree 0

knot contact homology:

Hcontact0 (K) ∼= Hstring

0 (K) ∼= Cord(K).

This recovers a result from the literature (see Theorem 1.1), modulo one impor-tant point. Recall (or see Section 6.2) that Hcontact

0 (K) is the degree 0 homology ofa differential graded algebra (A , ∂). In much of the literature on knot contact ho-mology, cf. [11, 32, 33], this DGA is an algebra over the coefficient ring Z[λ±1, µ±1]

(or Z[λ±1, µ±1, U±1], but in this paper we set U = 1): A is generated by a finitecollection of noncommuting generators (Reeb chords) along with powers of λ, µ thatcommute with Reeb chords. By contrast, in this paper (A , ∂) is the fully noncommu-tative DGA in which the coefficients λ, µ commute with each other but not with theReeb chords; see [12, 34].

The isomorphism Cord(K) ∼= Hcontact0 (K) in Theorem 1.1 is stated in the ex-

isting literature as an isomorphism of Z[λ±1, µ±1]-algebras, i.e., the coefficients λ, µcommute with everything for both Hcontact

0 (K) and Cord(K). However, an inspec-tion of the proof of Theorem 1.1 from [12, 31, 32] shows that it can be lifted to thefully noncommutative setting, in which λ, µ do not commute with Reeb chords (forHcontact

0 (K)) or cords (for Cord(K)). We omit the details here, and simply note thatour results give a direct proof of Theorem 1.1 in the fully noncommutative setting.

Remark 2.10. — Besides being more natural from the viewpoint of string homology,the stipulation that λ, µ do not commute with cords (in the cord algebra) or Reebchords (in the DGA) is essential for our construction, in Section 2.4 below, of a mapfrom degree 0 homology to the group ring of π, the fundamental group of the knotcomplement. This in turn is what allows us to (re)prove that knot contact homologydetects the unknot, among other things. If we pass to the quotient where λ, µ commutewith everything, then there is no well-defined map to Zπ.

Remark 2.11. — As already mentioned in the introduction, in [2] Basu, McGibbon,Sullivan and Sullivan have given a string topology description of a version of the cordalgebra for a codimension 2 submanifold K ⊂ Q of some ambient manifold Q, provinga theorem which formally looks quite similar to Proposition 2.9. In the language we

J.É.P. — M., 2017, tome 4


use here, the main difference in their work is the absence of N -strings, so that forknots K ⊂ R3 the version of Hstring

0 (K) they define only recovers the specializationat λ = 1 of (the commutative version of) Cord(K).

2.3. Homotopy formulation of the cord algebra. — We now reformulate the cordalgebra in terms of fundamental groups, more precisely the knot group and its pe-ripheral subgroup, along the lines of the Appendix to [31]. In light of Proposition 2.9,we will henceforth denote the cord algebra as Hstring

0 (K).We first introduce some notation. Let K be an oriented knot in an oriented 3-

manifold Q (in fact we only need an orientation and coorientation of K). Let Nbe a tubular neighborhood of K; as suggested by the notation, we will identify thisneighborhood with the conormal bundle N ⊂ T ∗Q via the tubular neighborhoodtheorem. We write

π = π1(QrK)

π = π1(∂N);

note that the inclusion ∂N → N induces a map π → π, typically an injection.Let Zπ, Zπ denote the group rings of π, π. We fix a framing on K; this, along withthe orientation and coorientation of K, allows us to specify two elements µ, λ for πcorresponding to the meridian and longitude, and to write

Zπ = Z[λ±1, µ±1].

The group ring Zπ and the cord algebra Hstring0 (K) both have natural maps from

Z[λ±1, µ±1] (which are injective unless K is the unknot). This motivates the followingdefinition, where “NC” stands for “noncommutative”.

Definition 2.12. — Let R be a ring. An R-NC-algebra is a ring S equipped with aring homomorphism R → S. Two R-NC-algebras S1, S2 are isomorphic if there is aring isomorphism S1 → S2 that commutes with the maps R→ S1, R→ S2.

Note that when R is commutative, the notion of an R-NC-algebra differs from theusual notion of an R-algebra; for example, an R-algebra S requires s1(rs2) = rs1s2

for r ∈ R and s1, s2 ∈ S, while an R-NC-algebra does not. (One can quotient anR-NC-algebra by commutators involving elements of R to obtain an R-algebra.) If Rand S are both commutative, however, then the notions agree. Also note that anyR-NC-algebra is automatically an R-bimodule, where R acts on the left and on theright by multiplication.

By the construction of the cord algebra Cord(K) from Section 2.2, Hstring0 (K) is a

Zπ-NC-algebra. We now give an alternative definition of Hstring0 (K) that uses π and π

in place of cords.A broken word in π, π is a nonempty word in elements of π and π whose letters

alternate between elements in π and π. For clarity, we use Roman letters for elementsin π and Greek for π, and enclose elements in π, π by square and curly brackets, respec-tively. Thus examples of broken words are α, [x], [x]α, and α1[x1]α2[x2]α3.

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Consider the Z-module freely generated by broken words in π, π, divided by thefollowing string relations:

(i) · · ·1 [xα1]α2 · · ·2 = · · ·1 [x]α1α2 · · ·2(ii) · · ·1 α1[α2x] · · ·2 = · · ·1 α1α2[x] · · ·2(iii) (· · ·1 [x1x2] · · ·2)− (· · ·1 [x1µx2] · · ·2) = · · ·1 [x1]1[x2] · · ·2(iv) (· · ·1 α1α2 · · ·2)− (· · ·1 α1µα2 · · ·2) = · · ·1 α1[1]α2 · · ·2.

Here · · ·1 is understood to represent the same (possibly empty) subword each time itappears, as is · · ·2. We denote the resulting quotient by S(π, π).

The Z-module S(π, π) splits into a direct sum corresponding to the four possiblebeginnings and endings for broken words:

S(π, π) = Sππ(π, π)⊕ Sππ(π, π)⊕ Sππ(π, π)⊕ Sππ(π, π),

where the superscripts denote which of π and π contain the first and last letters in thebroken word. Thus Sππ(π, π) is generated by broken words beginning and ending withcurly brackets (elements of π)— α, α1[x]α2, etc.—while Sππ(π, π) is generatedby [x], [x]α[y], etc. We think of these broken words as broken strings with basepoint on N ∩Q beginning and ending with N -strings (for Sππ(π, π)) or Q-strings (forSππ(π, π)). The other two summands Sππ(π, π), Sππ(π, π) can similarly be interpretedin terms of broken strings, but we will not consider them further.

On Sππ(π, π) and Sππ(π, π), we can define multiplications by

(· · ·1 α1)(α2 · · ·2) = · · ·1 α1α2 · · ·2(· · ·1 [x1])([x2] · · ·2) = · · ·1 [x1x2] · · ·2 ,and

respectively. These turn Sππ(π, π) and Sππ(π, π) into rings. Note for future referencethat Sππ(π, π) is generated as a ring by α and 1[x]1 for α ∈ π and x ∈ π.

Proposition 2.13. — Sππ(π, π) is a Zπ-NC-algebra, while Sππ(π, π) is a Zπ-NC-algebra and hence a Zπ-NC-algebra as well. Both Sππ(π, π) and Sππ(π, π) are knotinvariants as NC-algebras.

Proof. — We only need to specify the ring homomorphisms Zπ → Sππ(π, π) andZπ → Sππ(π, π); these are given by α 7→ α and x 7→ [x], respectively.

Remark 2.14. — View Zπ as a Zπ-bimodule via the map π → π. Then Sππ(π, π) andSππ(π, π) can alternatively be defined as follows. Let A , A be defined by

A = Zπ ⊕ Zπ ⊕ (Zπ ⊗Zπ Zπ)⊕ (Zπ ⊗Zπ Zπ ⊗Zπ Zπ)⊕ · · ·

A = Zπ ⊕ (Zπ ⊗Zπ Zπ)⊕ (Zπ ⊗Zπ Zπ ⊗Zπ Zπ)⊕ · · · .

Each of A , A has a multiplication operation given by concatenation (e.g. a · (b⊗ c) =

a ⊗ b ⊗ c); multiplying by an element of Zπ ⊂ A uses the Zπ-bimodule structureon Zπ. There are two-sided ideals I ⊂ A , I ⊂ A generated by

x1x2 − x1µx2 − x1 ⊗ x2

1π − (1− µ)π

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where x1, x2 ∈ π, x1x2, x1µx2 are viewed as elements in Zπ, and 1π denotes theelement 1 ∈ Zπ while (1− µ)π denotes the element 1− µ ∈ Zπ. Then

Sππ(π, π) ∼= A /I

Sππ(π, π) ∼= A /I .

We conclude this subsection by noting that Sππ(π, π) is precisely the cord algebraof K.

Proposition 2.15. — We have the following isomorphism of Zπ-NC-algebras:

Hstring0 (K) ∼= Sππ(π, π).

Proof. — We use the cord-algebra formulation of Hstring0 (K) ∼= Cord(K) from Defini-

tion 2.6. LetK ′ be the parallel copy ofK, and choose a base point p′ for π = π1(QrK)

with p′ ∈ K ′ r ∗. Given a cord γ of K, define γ ∈ π as in Remark 2.8: extend γto a closed loop γ in Q r K with endpoints at p′ by connecting the endpoints of γto p′ along K ′ r ∗. Then the isomorphism φ : Cord(K) → Sππ(π, π) is the ringhomomorphism defined by:

φ(γ) = 1[γ]1φ(α) = α,

for γ any cord of K and α any element of Cord(K) of the form λaµb.The skein relations in Cord(K) from Definition 2.6 are mapped by φ to:(i) 1[1]1 = 1 − µ(ii) 1[µγ]1 = µ[γ]1 and 1[γµ]1 = 1[γ]µ(iii) 1[λγ]1 = λ[γ]1 and 1[γλ]1 = 1[γ]λ(iv) 1[γ1γ2]1 − 1[γ1µγ2]1 = 1[γ1]1[γ2]1.

In Sππ(π, π), these follow from string relations (iv), (i) and (ii), (i) and (ii), and (iii),respectively.

Thus φ is well-defined. It is straightforward to check that φ is an isomorphism(indeed, the string relations are constructed so that this is the case), with inverse φ−1

defined byφ−1(α) = α

φ−1(1[γ]1) = γ,

for α ∈ π and γ ∈ π: note that a closed loop at p′ ∈ K ′ r ∗ representing γ is alsoby definition a cord of K.

Remark 2.16. — Similarly, one can show that Hstring0 (K) ∼= Sππ(π, π) as Zπ-NC-

algebras. In the same vein, there is also a cord formulation for modified string homo-logy Hstring

0 (K) (as introduced at the end of Section 2.1), along the lines of Defini-tion 2.6: this is A /I , where A is the non-unital algebra generated by nonemptyproducts of cords (the difference from A being that A does not contain words ofthe form λaµb, which have no cords), and I is the ideal of A generated by skeinrelations (ii) through (iv) from Definition 2.6, without (i).

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2.4. The cord algebra and group rings. — Having defined the cord algebra in termsof homotopy groups, we can now give an even more explicit interpretation not involv-ing broken words, in terms of the group ring Zπ. Notation is as in Section 2.3: in partic-ular, K ⊂ Q is a framed oriented knot with tubular neighborhood N , π = π1(QrK),and π = π1(∂N). When Q = R3, we assume for simplicity that the framing on K isthe Seifert framing.

Before addressing the cord algebra Hstring0 (K) ∼= Sππ(π, π) itself, we first note that

the modified version Hstring0 (K) ∼= Sππ(π, π) is precisely Zπ.

Proposition 2.17. — For a knot K ⊂ Q, we have an isomorphism as Zπ-NC-algebras

Sππ(π, π) ∼= Zπ.

Proof. — The map Zπ → Sππ(π, π), x 7→ [x], has inverse φ given by

φ([x1]α1[x2]α2 · · · [xn−1]αn−1[xn])

= x1(1− µ)α1x2(1− µ)α2 · · ·xn−1(1− µ)αn−1xn;

note that φ is well-defined (just check the string relations) and preserves ring structure.

The corresponding description of the cord algebra Sππ(π, π) is a bit more involved,and we give two interpretations.

Proposition 2.18. — For a knot K ⊂ Q, we have a Z-module isomorphism

Sππ(π, π) ∼= Z[λ±1]⊕ Zπ.

For any α ∈ Z[λ±1], the left and right actions of α on Sππ(π, π) induced from the π-NC-algebra structure on Sππ(π, π) coincide under this isomorphism with the actionsof α on the factors of Z[λ±1]⊕ Zπ by left and right multiplication.

Proof. — The isomorphism Z[λ±1]⊕Zπ → Sππ(π, π) sends (α, 0) to α and (0, x) to1[x]1. Note that this map commutes with left and right multiplication by powersof λ; for example, λkα = λkα and 1[λkx]1 = λk[x]1 = λk1[x]1.

To see that the map is a bijection, note that the generators of Sππ(π, π) can beseparated into “trivial” broken words of the form α and “nontrivial” broken wordsof length at least 3. Using the string relations, we can write any trivial broken worduniquely as a sum of some λa and some nontrivial broken words:

λaµb = λa −b−1∑i=0

λa[µi−1]1

if b > 0, and similarly for b < 0. On the other hand, any nontrivial broken word inSππ(π, π) can be written uniquely as a Z-linear combination of words of the form1[x]1, x ∈ π: just use the map φ from the proof of Proposition 2.17 to reduceany nontrivial broken word to broken words of length 3, and then apply the identityα1[x]α2 = 1[α1xα2]1.

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Proposition 2.19. — For knots K ⊂ R3, string homology Hstring0 (K) ∼= Sππ(π, π),

and thus knot contact homology, detects the unknot U . More precisely, left multipli-cation by λ− 1 on Sππ(π, π) has nontrivial kernel if and only if K is unknotted.

Proof. — First, if K = U , then λ = 1 in π, and so

(λ− 1)1[1]1 = λ[1]1 − 1[1]1 = 1[λ]1 − 1[1]1 = 0

in Hstring0 (U), while 1[1]1 6= 0 by the proof of Proposition 2.18.

Next assume that K 6= U , and consider the effect of multiplication by λ − 1 onZ[λ±1] ⊕ Zπ. Clearly this map is injective on the Z[λ±1] summand; we claim thatit is injective on the Zπ summand as well. Indeed, suppose that some nontrivialsum

∑ai[xi] ∈ Zπ is unchanged by multiplication by λ. Then [λkx1] must appear

in this sum for all k, whence the sum is infinite since π injects into π by the LoopTheorem.

Remark 2.20. — It was first shown in [32] that the cord algebra detects the unknot.That proof uses a relationship between the cord algebra and the A-polynomial, alongwith the fact that the A-polynomial detects the unknot [8], which in turn relies ongauge-theoretic results of Kronheimer and Mrowka [28]. As noted previously, by con-trast, the above proof that string homology detects the unknot uses only the LoopTheorem. Either proof shows that knot contact homology detects the unknot. How-ever, we emphasize that for our argument, unlike the argument in [32], it is crucialthat we use the fully noncommutative version of knot contact homology.

We can recover the multiplicative structure on Sππ(π, π) under the isomorphismof Proposition 2.18 as follows. On Z[λ±]⊕Zπ, define a multiplication operation ∗ by

(λk1 , x1) ∗ (λk2 , x2) = (λk1+k2 , λk1x2 + x1λk2 + x1x2 − x1µx2).

It is easy to check that ∗ is associative, and that the isomorphism

Sππ(π, π) ∼= (Z[λ±]⊕ Zπ, ∗)

now becomes an isomorphism of Zπ-NC-algebras, where Z[λ±1] ⊕ Zπ is viewed asa Zπ-NC-algebra via the map Z[π] → Z[λ±1] ⊕ Zπ sending λ to (λ, 0) and µ to(1, 0)− (0, 1).

We now turn to another formulation of string homology in terms of the groupring Zπ. This formulation is a bit cleaner than the one in Proposition 2.18, as themultiplication operation is easier to describe.

Proposition 2.21. — For a knot K ⊂ Q, let R denote the subring of Zπ generatedby π and im(1−µ), where 1−µ denotes the map Zπ → Zπ given by left multiplicationby 1− µ. There is a ring homomorphism

ψ : Hstring0 (K) −→ R

determined by ψ(α) = α and ψ(1[x]1) = x− µx.If π → π is an injection (in particular, if K ⊂ R3 is nontrivial), then ψ is an

isomorphism of Zπ-NC-algebras.

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Proof. — It is easy to check that ψ respects all of the string relations definingSππ(π, π) ∼= Hstring

0 (K): the key relation [x1x2] − [x1µx2] − [x1]1[x2] is sent to(1−µ)x1x2− (1−µ)x1µx2− (1−µ)x1(1−µ)x2 = 0. Thus ψ is well-defined as a mapHstring

0 (K) → R. This map acts as the identity on π and thus is a Zπ-NC-algebramap.

Since ψ is surjective by construction, it remains only to show that ψ is injectivewhen π → π is injective. Suppose that

(2.1) 0 = ψ

(∑i

aiλi+∑j

bj1[xj ]1)

=∑i

aiλi +∑j

bj(1− µ)xj

for some ai, bj ∈ Z and xj ∈ π. We claim that bj = 0 for all j, whence ai = 0 forall i since π injects into π for K nontrivial. Assume without loss of generality thatthe framing on K is the 0-framing (changing framing simply replaces λ by λµk forsome k). Then the linking number withK gives a homomorphism lk : π → Z satisfyinglk(λ) = 0 and lk(µ) = 1. If

∑j bjxj is not a trivial sum, then let x` be the contributor

to this sum of maximal linking number. The term −b`µx` in∑j bj(1−µ)xj cannot be

canceled by any other term in that sum; thus for (2.1) to hold, x` must have linkingnumber −1. But a similar argument shows that the contributor to

∑j bjxj of minimal

linking number must have linking number 0, contradiction. We conclude that∑j bjxj

must be a trivial sum, as claimed.

Remark 2.22. — To be clear, as a knot invariant derived from knot contact homology,the cord algebra Hstring

0 (K) (for K 6= U) is the ring R ⊂ Zπ along with the mapZπ = Z[λ±1, µ±1] → R. Proposition 2.21 implies that the Z[λ±1, µ±1]-NC-algebrastructure on Zπ = Hstring

0 (K) completely determines Hstring0 (K). We do not know if

Hstring0 (K) determines Hstring

0 (K) as well, nor whether Hstring0 (K) is a complete knot

invariant.(4)

On the other hand, Hstring0 (K) as a ring is a complete knot invariant for prime knots

in R3 up to mirroring, as we can see as follows. By Proposition 2.17, Hstring0 (K) ∼=

Z[π], and for prime knots K, Gordon and Luecke [24] show that π = π1(R3 r K)

determines K up to mirroring. On the other hand, π is a left-orderable group, andthe ring isomorphism type of Z[G] when G is left-orderable is determined by the groupisomorphism type of G [29]. We thank Tye Lidman for pointing this out to us.

We conclude this section with two examples.

Example 2.23. — When K is the unknot U , then

Hstring0 (U) ∼= Z[λ±1, µ±1]/((λ− 1)(µ− 1)),

(4)Added in revision: it has now been proven by Shende [35], and then reproven in [17], thatthe Legendrian isotopy type of ΛK completely determines the knot K. The proof in [17] relies onthe present paper and shows that an enhanced version of knot contact homology (or of Hstring

0 (K))determines K. The question of whether Hstring

0 (K) is a complete invariant remains open.

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while Hstring0 (U) ∼= Z[µ±1]. The ring homomorphism ψ from Proposition 2.21, which is

not injective, is given by ψ(λ) = 1, ψ(µ) = µ. The isomorphism from Proposition 2.18is the (inverse of the) map

Z[λ±1]⊕ Z[µ±1] −→ Z[λ±1, µ±1]/((λ− 1)(µ− 1))

(α, β) 7−→ α+ (µ− 1)β.

As noticed by Lidman, this computation of Hstring0 (U) along with Proposition 2.21

gives an alternative (and shorter) proof that knot contact homology detects the unknot(Proposition 2.19), and more generally that this continues to hold even if the knot isnot assumed to be Seifert framed (Corollary 1.5).

Proof of Corollary 1.5. — Suppose that Hstring0 (K) ∼= Hstring

0 (U) where K is a framedoriented knot and U is the unknot with some framing. By changing the framing ofboth, we can assume that K has its Seifert framing. If K is knotted, then Zπ hasno zero divisors since π is left-orderable, and thus Hstring

0 (K) ⊂ Zπ also has no zerodivisors by Proposition 2.21. On the other hand,

Hstring0 (U) ∼= Z[λ±1, µ±1]/((λµf − 1)(µ− 1))

for some f ∈ Z. Thus K must be the unknot and must further have the same framingas U .

In [23], Gordon and Lidman extend this line of argument (i.e., applying Proposi-tion 2.21) to prove that knot contact homology detects torus knots as well as cablingand compositeness.

Example 2.24. — When K is the right-handed trefoil T , a slightly more elaborateversion of the calculation of the cord algebra from [32] (see also [34]) gives the followingexpression forHstring

0 (T ): it is generated by λ±1, µ±1, and one more generator x, alongwith the relations:

λµ = µλ

λµ6x = xλµ6

−1 + µ+ x− λµ5xµ−3xµ−1 = 0

1− µ− λµ4xµ−2 − λµ5xµ−2xµ−1 = 0.

On the other hand, Hstring0 (T ) = Zπ is the ring generated by µ±1 and a±1 modulo

the relation µaµ = aµa; the longitudinal class is λ = aµa−1µaµ−3. The explicit mapfrom Hstring

0 (T ) to Zπ is given by:

µ 7−→ µ

λ 7−→ λ = aµa−1µaµ−3

x 7−→ (1− µ)aµ−1a−1.

It can be checked that this map preserves the relations in Hstring0 (T ).

J.É.P. — M., 2017, tome 4


3. Roadmap to the proof of Theorem 1.2

The remainder of this paper is devoted to the proof of Theorem 1.2. To avoidgetting lost in the details, we give here a roadmap to the proof and explain thetechnical issues to be addressed along the way.

The proof follows the scheme that is described for a different situation in [6] andconsists of 3 steps. Let A be the free Zπ-NC-algebra generated by Reeb chords and∂Λ : A → A the boundary operator for Legendrian contact homology. For a Reebchord a and an integer ` > 0 denote by M`(a) the moduli space of J-holomorphicdisks in T ∗Q with one positive puncture asymptotic to a and boundary on Q ∪ LKwith 2` corners at which it switches between LK and Q.

Step 1. — Show that M`(a) can be compactified to a manifold with corners M `(a)

and that the generating functions φ(a) :=∑∞`=0 M `(a) (extended as algebra maps

to A ) satisfy the relation∂φ = φ∂Λ − δφ,

where δM `(a) is the subset of elements in M `(a) that intersect K at the interior ofsome boundary string.

Step 2. — Construct a chain complex (C∗(Σ), ∂ + δ) of suitable chains of brokenstrings such that φ induces a chain map

Φ : (A , ∂Λ) −→ (C(Σ), ∂ + δ),

and the homology H0(Σ, ∂+δ) agrees with the string homology Hstring0 (K) as defined

in Section 2.1.

Step 3. — Prove that Φ induces an isomorphism on homology in degree zero.

Step 1 occupies Sections 8 to 10. It involves detailed descriptions of– the behavior of holomorphic disks at corner points;– compactifications of moduli spaces of holomorphic disks;– transversality and gluing of moduli spaces.In Step 2 (Sections 4 to 6) we encounter the following problem: The direct approach

to setting up the complex (C(Σ), ∂ + δ) would involve chains in spaces of brokenstrings with varying number of switches. These spaces could probably be given smoothstructures using the polyfold theory by Hofer, Wysocki and Zehnder [27]. Here wechoose a different approach, keeping the number of switches fixed and inserting small“spikes” in the definition of the string operation δ = δQ + δN . Since this involvesnon-canonical choices, one does not expect identities such as ∂δ + δ∂ = 0 to holdstrictly but only up to homotopy, thus leading to an ∞-structure as described bySullivan in [37]. We avoid ∞-structures by carefully defining δ via induction over thedimension of chains such that all identities hold strictly on the chain level.

J.É.P. — M., 2017, tome 4


Step 3 (Section 7) follows the scheme described in [6]. This involves– a length estimate for the boundary of holomorphic disks, which implies that Φ

respects the filtrations of A and C(Σ) by the actions of Reeb chords and the totallengths of Q-strings, respectively.

– construction of a length-decreasing chain homotopy deforming C(Σ) to chainsC(Σlin) of broken strings all of whose Q-strings are linear straight line segments(at this point we specialize to Q = R3);

– Morse-theoretical arguments on the space Σlin to prove that Φ induces an iso-morphism on degree zero homology.

4. Holomorphic functions near corners

In this section, we call a function f : R → C on a subset R ⊂ C with piecewisesmooth boundary holomorphic if it is continuous on R and holomorphic in the interiorof R.

4.1. Power series expansions. — Denote by D ⊂ C the open unit disk and set

D+ := z ∈ D | =(z) > 0,Q+ := z ∈ D | <(z) > 0, =(z) > 0.

Consider a holomorphic function f : Q+ → C (in the above sense, i.e., continuouson Q+ and holomorphic in the interior) with f(0) = 0. We distinguish four casesaccording to their boundary conditions.

Case 1. — f maps R+ to R and iR+ to iR.In this case, we extend f to a map f : D+ → C by the formula

f(z) := −f(−z), <(z) 6 0, =(z) > 0,

and then to a map f : D → C by the formula

f(z) := f(z), =(z) 6 0.

The resulting map f is continuous on D and holomorphic outside the axes R ∪ iR,hence holomorphic on D, and it maps R to R and iR to iR. Thus it has a power seriesexpansion

f(z) =

∞∑j=1

a2j−1z2j−1, aj ∈ R.

This shows that each holomorphic function f : Q+ → C mapping R+ to R and iR+

to iR is uniquely the restriction of such a power series. In particular, f has an isolatedzero at the origin unless it vanishes identically. Similar discussions apply in the othercases.

Case 2. — f maps (R+, iR+) to (iR,R). Then it has a power series expansion

f(z) = i

∞∑j=1

a2j−1z2j−1, aj ∈ R.

J.É.P. — M., 2017, tome 4


Case 3. — f maps (R+, iR+) to (R,R). Then it has a power series expansion

f(z) =

∞∑j=1

a2jz2j , aj ∈ R.

Case 4. — f maps (R+, iR+) to (iR, iR). Then it has a power series expansion

f(z) = i

∞∑j=1

a2jz2j , aj ∈ R.

Remark 4.1. — We can summarize the four cases by saying that f : Q+ → C is givenby a power series

f(z) =

∞∑k=1

akzk

with either only odd (in Cases 1 and 2) or only even (in Cases 3 and 4) indices k, andwith the ak either all real (in Cases 1 and 3) or all imaginary (in Cases 2 and 4). Suchholomorphic functions f will appear as projections onto a normal direction of theholomorphic curves considered in Section 6.3 near switches. Then Case 1 correspondsto a switch from Q to N , Case 2 to a switch from N to Q, Case 3 to a switch from N

to N , and Case 4 to a switch from Q to Q.

Remark 4.2. — It will sometimes be convenient to switch from the positive quadrantto other domains. For example, the map ψ(z) :=

√z maps the upper half disk D+

biholomorphically onto Q+. Thus in Case 1 the composition f ψ is a holomorphicfunction on D+ which maps R+ to R and R− to iR, and it has an expansion in powersof√z by

f ψ(z) =

∞∑j=1

a2j−1zj−1/2, aj ∈ R.

As another example, the map φ(s, t) := ie−π(s+it)/2 maps the strip (0,∞) × [0, 1]

biholomorphically onto Q+r0. Thus in Case 1 the composition f φ is a continuousfunction on R+× [0, 1] which is holomorphic in the interior and maps R+×0 to iRand R+ × 1 to R, and it has a power series expansion

f φ(s, t) = −i∞∑j=1

(−1)ja2j−1e−(2j−1)π(s+it)/2, aj ∈ R.

Similar discussions apply to the other cases.

Let us consider once more the function f : Q+ → C of Case 1 mapping (R+, iR+)

to (R, iR). Its restrictions to iR+ resp. R+ naturally give rise to functions

f− : (−1, 0] −→ R resp. f+ : [0, 1) −→ R

viaf−(t) := (−i)f(−it), t 6 0, f+(t) := f(t), t > 0.

J.É.P. — M., 2017, tome 4


Here and in the sequel we always use the isomorphism (−i) = i−1 : iR→ R to iden-tify iR with R in the target. So f± are related by f− = r∗f+, where the reflection r∗fof a complex valued power series f(t) =

∑∞k=1 akt

k, ak ∈ Cn, is defined by

r∗f(t) := (−i)f(−it) =

∞∑k=1

(−i)k+1aktk.

(Note that the domain C and the target C play different roles here: multiplication by(−i) on the domain comes from opening up the positive quadrant to the upper halfplane, while multiplication by (−i) in the target corresponds to the canonical rotationby −J from iR ⊂ Q to R ⊂ N .)

The effect of r∗ on the power series expansion f(t) =∑∞j=1 a2j−1t

2j−1 in Case 1 isas follows:

r∗f(t) = (−i)∞∑j=1

a2j−1(−it)2j−1 =

∞∑j=1

(−1)ja2j−1t2j−1,

so the coefficient a2j−1 is changed to (−1)ja2j−1. Note that a1 is changed to −a1,which justifies the name “reflection”.

Now consider f as in Case 2 mapping (R+, iR+) to (iR,R). Here the restrictionsto iR+ resp. R+ naturally give rise to functions f− : (−1, 0]→ R resp. f+ : [0, 1)→ Rvia

f−(t) := f(−it), t 6 0, f+(t) := (−i)f(t), t > 0.

So f± are related by f− = −r∗f+, and the coefficient a2j−1 in the power seriesexpansion of f+ is changed to (−1)j+1a2j−1. In particular, a1 is unchanged so that f−and f+ fit together to a function (−1, 1)→ R of class C2 (but not C3).

4.2. Winding numbers. — Consider a holomorphic function f : Q+ → C given by apower series f(z) =

∑∞k=1 akz

k as in Cases 1–4 of the previous subsection. In each ofthese cases we define its winding number at 0 as

w(f, 0) :=1

2infk | ak 6= 0.

Note that the winding number is a half-integer in the first two cases and an integerin the last two cases. Also note that the winding number is given by

w(f, 0) =1

π

∫γ

f∗dθ,

where γ is a small arc in Q+ connecting (0, 1) to i(0, 1). This can be seen, for example,by choosing γ as a small quarter circle Q+∩∂Dε; then the symmetry of f with respectto reflections at the coordinate axes implies

1

π

∫γ

f∗dθ =1

4π

∫∂Dε

f∗dθ =1

4π· 2π infk | ak 6= 0 = w(f, 0).

Next let r > 1, denote by Dr the open disk of radius r, by

H+ := z ∈ C | =(z) > 0

J.É.P. — M., 2017, tome 4


the upper half plane, and set D+r := Dr ∩ H+. Consider a nonconstant continuous

map f : D+r → C which is holomorphic in the interior and maps the interval (−r, r)

to R ∪ iR. Suppose that f has no zeroes on the semi-circle ∂D1 ∩ H+. Then f hasfinitely many zeroes s1, . . . , sk in the interior of D+

1 as well as finitely many zeroest1, . . . , t` in (−1, 1). (Finiteness holds because the holomorphic function z 7→ f(z)2

maps R to R, and thus can only have finitely many zeroes by the Schwarz reflectionprinciple and unique continuation.) Denote by w(f, si) ∈ N resp. w(f, tj) ∈ 1

2N thewinding numbers at the zeroes. Thus with the closed angular form dθ on Cr 0,

w(f, si) :=1

π

∫αi

f∗dθ, w(f, tj) :=1

π

∫βj

f∗dθ,

where αi is a small circle around si and βj is a small semi-circle around tj in D+1 ,

both oriented in the counterclockwise direction. (Thus the w(f, si) are even integersand the w(f, tj) are integers or half-integers). Denote by γ the semi-circle ∂D1 ∩H+

oriented in the counterclockwise direction. Then Stokes’ theorem yields

1

π

∫γ

f∗dθ =

k∑i=1

w(f, si) +∑j=1

w(f, tj).

Since all winding numbers are nonnegative, we have shown the following result.

Lemma 4.3. — Consider a nonconstant continuous map f : D+r → C which is holo-

morphic in the interior and maps (−r, r) to R ∪ iR. Suppose that f has no zeroes onthe semi-circle γ = ∂D1 ∩H+ and zeroes at t1, . . . , tm ∈ (−1, 1) (plus possibly furtherzeroes in D+

1 ). Then1

π

∫γ

f∗dθ >m∑j=1

w(f, tj).

More generally, for n > 1 consider a nonconstant continuous map f : D+r → Cn

which is holomorphic in the interior and maps (−r, r) to Rn∪iRn. Suppose that f hasno zeroes on the semi-circle ∂D1 ∩H+ and zeroes z1, . . . , zm in D+

1 (in the interior oron the boundary). For each direction v ∈ Sn−1 ⊂ Rn we obtain a holomorphic mapfv := πv f , where πv is the projection onto the complex line spanned by v. Fix apositive volume form Ω on Sn−1 of total volume 1. Then there exists an open subsetV ⊂ Sn−1 of measure 1 such that for all v ∈ V , fv has zeroes precisely at the zj andtheir winding numbers are independent of v ∈ V . So we can define

w(f, zj) :=

∫V

w(fv, zj)Ω(v) = w(fv0 , zj)

for any v0 ∈ V and obtain

Corollary 4.4. — Consider a nonconstant continuous map f : D+r → Cn which is

holomorphic in the interior and maps (−r, r) to Rn∪iRn. Suppose that f has no zeroeson the semi-circle γ = ∂D1 ∩ H+ and zeroes at t1, . . . , tm ∈ (−1, 1) (plus possibly

J.É.P. — M., 2017, tome 4


fε

fδ,ε

0 ε

0 ε−δ

Figure 4.1. Spikes in the model families fε and fδ,ε.

further zeroes in D+1 ). Then there exists an open subset V ⊂ Sn−1 of measure 1 such

that for every v0 ∈ V ,1

π

∫γ

f∗v0dθ =

∫V

(1

π

∫γ

f∗v dθ

)Ω(v) >

m∑j=1

w(f, tj).

4.3. Spikes. — Consider again the upper half disk D+ = z ∈ D | =(z) > 0 andreal points −1 < b1 < b2 < · · · < b` < 1. We are interested in holomorphic functionsf : D+ rb1, . . . , b`, continuous on D+, mapping the intervals [bi−1, bi] alternatinglyto R and iR. We wish to describe models of 1- resp. 2-parameter families in which 2resp. 3 of the bi come together. A model for such a 1-parameter family is

(4.1) fε(z) :=√z(z − ε), ε > 0

with zeroes at 0, ε. A model for a 2-parameter family is

(4.2) fδ,ε(z) :=√z(z + δ)(z − ε), ε, δ > 0

with zeroes at −δ, 0, ε. Here we choose appropriate branches of the square root sothat the functions become continuous. The images of these functions are shown inFigure 4.1. They show that fε has a “spike” in the direction iR+ which disappearsas ε → 0, and fδ,ε has two “spikes” in the directions R− resp. iR+ which disappearas δ resp. ε approaches zero. Based on these models, the notion of a “spike” will beformalized in Section 5.

J.É.P. — M., 2017, tome 4


In the following section, functions with two spikes will appear in the following localmodel. Consider the 1-parameter family of functions fa : Q+ → C,

fa(z) = i(az − z3), a ∈ R.

They map (R+, iR+) to (iR,R) and thus correspond to Case 2 in Section 4.1. Via theidentifications in that section, fa induces functions

f−(a, t) := fa(−it) = at+ t3, t 6= 0,

f+(a, t) := (−i)fa(t) = at− t3, t > 0,

which fit together to a C2 (though not C3) function

f(a, t) = at− sgn(t)t3, t ∈ R.

In Case 1, one considers the functions fa(z) = −az+z3 mapping (R+, iR+) to (R, iR).Here the induced functions

f−(a, t) := (−i)fa(−it) = at+ t3, t 6= 0,

f+(a, t) := fa(t) = −at+ t3, t > 0

do not fit together to a C1 function, but when we replace f+ by −f+ they fit togetherto the function f(a, t) above.

5. String homology in arbitrary degree

5.1. Broken strings. — Let K be a framed oriented knot in some oriented 3-man-ifold Q. Fix a tubular neighborhood N of K and a diffeomorphism N ∼= S1 ×D2.

Fix an integer m > 3 and a base point x0 ∈ ∂N . We also fix an m-jet of a curvepassing through x0 in N . Using the diffeomorphism N ∼= S1 ×D2, this is equivalentto specifying suitable vectors v(k)

0 ∈ R3, 1 6 k 6 m. The following definition refinesthe one given in Section 2, which corresponds to the case m = 1.

Definition 5.1. — A broken (closed) string with 2` switches on K is a tuple s =

(a1, . . . , a2`+1; s1, . . . , s2`+1) consisting of real numbers 0 = a0 < a1 < · · · < a2`+1

and Cm-maps

s2i+1 : [a2i, a2i+1] −→ N, s2i : [a2i−1, a2i] −→ Q

satisfying the following matching conditions at the end points ai:(i) s1(0) = s2`+1(a2`+1) = x0 and s(k)

1 (0) = s(k)2`+1(a2`+1) = v

(k)0 for 1 6 k 6 m.

(ii) For i = 1, . . . , `,

s2i(a2i) = s2i+1(a2i) ∈ K, s2i−1(a2i−1) = s2i(a2i−1) ∈ K.

(iii) Denote by σi the D2-component of si near its end points. Then for i = 1, . . . , `

and 1 6 k 6 m/2 (for the left hand side) resp. 1 6 k 6 (m+ 1)/2 (for the right handside)

σ(2k)2i (a2i) = σ

(2k)2i+1(a2i) = 0, σ

(2k−1)2i (a2i) = (−1)kσ

(2k−1)2i+1 (a2i),

σ(2k)2i−1(a2i−1) = σ

(2k)2i (a2i−1) = 0, σ

(2k−1)2i−1 (a2i−1) = (−1)k+1σ

(2k−1)2i (a2i−1).

J.É.P. — M., 2017, tome 4


We will refer to the s2i and s2i+1 as Q-strings and N-strings, respectively. A typicalpicture of a broken string is shown in Figure 2.1 on page 666. Conditions (i) and (ii)in Definition 5.1 mean that the si fit together to a continuous loop s : [0, a2`+1]→ Q

with end points at x0 (which fit together in Cm).Condition (iii) is motivated as follows: In Section 8 below, we consider almost

complex structures J on T ∗Q which are particularly well adapted to the immersedLagrangian submanifold Q ∪ LK ⊂ T ∗Q. For such a J , Lemma 8.6 then provides aholomorphic embedding of a neighborhood O of S1×0 ⊂ C×C2 onto a neighborhoodof K ⊂ T ∗Q mapping O ∩ (S1 × iR2) to Q and O ∩ (S1 ×R2) to LK . Condition (iii)requires that the normal component σ of s at the switching points ai behaves likethe boundary values of a holomorphic disk with boundary on Q∪LK when projectedto C2 in these coordinates near K.

To see this, let us reformulate condition (iii). As in Section 4.1, to a complex valuedpolynomial p(t) =

∑mk=1 pkt

k, pk ∈ C2, we associate its reflection

r∗p(t) = (−i)p(−it) =

m∑k=1

(−i)k+1pktk.

Then two real valued polynomials p(t) =∑mk=1 pkt

k and q(t) =∑mk=1 qkt

k,pk, qk ∈ R2, satisfy r∗p = q if and only if for 1 6 k 6 m/2 (on the left hand side)resp. 1 6 k 6 (m+ 1)/2 (on the right hand side)

p2k = q2k = 0 and p2k−1 = (−1)kq2k−1.

So in terms of the normal Taylor polynomials at the switching points

Tmσi(ai−1)(t) :=

m∑k=1

σ(k)i (ai−1)

k!tk, Tmσi(ai)(t) :=

m∑k=1

σ(k)i (ai)

k!tk,

condition (iii) is equivalent to the conditions

Tmσ2i(a2i) = r∗Tmσ2i+1(a2i), Tmσ2i−1(a2i−1) = −r∗Tmσ2i(a2i−1).

These are precisely the conditions in Section 4.1 describing the boundary behaviorof holomorphic disks at a corner going from the imaginary to the real axis (Case 1,corresponding to a switch from Q to N), resp. from the real to the imaginary axis(Case 2, corresponding to a switch from N to Q).

Remark 5.2(a) The case m = 3 suffices for the purposes of this paper. In fact, for 0- and

1-parametric families of strings we only need the conditions on the first derivatives(the case m = 1 considered in Section 2), while for 2-parametric families we alsoneed the conditions on the second and third derivatives). Explicitly, condition (iii) form = 3 reads

(5.1)σ′2i(a2i) = −σ′2i+1(a2i), σ′2i−1(a2i−1) = σ′2i(a2i−1),

σ′′2i(a2i) = σ′′2i+1(a2i) = σ′′2i−1(a2i−1) = σ′′2i(a2i−1) = 0,

σ′′′2i(a2i) = σ′′′2i+1(a2i), σ′′′2i−1(a2i−1) = −σ′′′2i(a2i−1).

J.É.P. — M., 2017, tome 4


(b) In Definition 5.1 one could add the condition that all derivatives of the tangentcomponents agree at switches (as it is the case for boundaries of holomorphic disks).However, we will not need such a condition and thus chose not to include it. Similarly,one could have required all the sj to be C∞ rather than Cm.

We denote by Σ` the space of broken strings with 2` switches. We make it a Banachmanifold by equipping it with the topology of R on the aj and the Cm-topology onthe sj . It comes with interior evaluation maps

evi : (0, 1)× Σ` −→ Q resp. N, (t, s) 7−→ si((1− t)ai−1 + tai

)and corner evaluation maps

Ti : Σ` −→ (R2)b(m+1)/2c, s 7−→ Tmσi(ai) ∼=(σ

(2k−1)i (ai)

)16k6b(m+1)/2c.

Moreover, concatenation at the base point x0 yields a smooth map

Σ` × Σ`′7−→ Σ`+`

′.

5.2. Generic chains of broken strings. — Now we define the generators of the stringchain complex in degrees d ∈ 0, 1, 2. Set ∆0 := 0 and let

∆d = (λ1, . . . , λd) ∈ Rd | λi > 0, λ1 + · · ·+ λd 6 1

denote the d-dimensional standard simplex for d > 1. It is stratified by the sets wheresome of the inequalities are equalities. Fix m > 3 as in the previous subsection.

Definition 5.3. — A generic d-chain in Σ` is a smooth map S : ∆d → Σ` such thatthe maps evi S : (0, 1) × ∆d → Q and Ti S : ∆d → (R2)b(m+1)/2c are jointlytransverse to K resp. jet-transverse to 0 (on all strata of ∆d).

Let us spell out what this means for m = 3 in the cases d = 0, 1, 2.

d = 0. — A generic 0-chain is a broken string s = (s1, . . . , s2`+1) such that(0a) σi(ai) 6= 0 for all i;(0b) si intersects K only at its end points.

d = 1. — A generic 1-chain of broken strings is a smooth map

S : [0, 1] −→ Σ`, λ 7−→ sλ = (sλ1 , . . . , sλ2`+1)

such that(1a) s0 and s1 are generic strings;(1b) σλi (aλi ) 6= 0 for all i, λ;(1c) for each i the map

(0, 1)× (0, 1) −→ Q resp. N, (t, λ) −→ sλi((1− t)aλi−1 + taλi

)meets K transversely in finitely many points (ta, λa). Moreover, distinct such inter-sections (even for different i) appear at distinct parameter values λa.

J.É.P. — M., 2017, tome 4


q1

p1

Figure 5.1. A spike with ends (p, q).

d = 2. — A generic 2-chain of broken strings is a smooth map

S : ∆2 −→ Σ`, λ 7−→ sλ = (sλ1 , . . . , sλ2`+1)

such that(2a) the sλ at vertices λ ∈ ∆2 are generic strings;(2b) the restrictions of S to edges of ∆2 are generic 1-chains;(2c) for each i the map

(0, 1)× int∆2 −→ Q resp. N, (t, λ) −→ sλi((1− t)aλi−1 + taλi

)is transverse to K; moreover, we assume that the projection of the preimage of Kto ∆2 is an immersed submanifold Di ⊂ ∆2 with transverse double points;

(2d) for all i, j the submanifolds Di, Dj ⊂ ∆2 from(2c) meet transversely in finitely many points;(2e) for each i the map

int∆2 −→ R2, λ 7−→ σλi (aλi )

meets 0 transversely in finitely many points satisfying (σλi )(3)(aλi ) 6= 0; moreover,these points do not meet the Dj .

We will see in the next subsection that the points in (2e) are limit points of both Di

and Di+1.

5.3. String operations. — Now we define the relevant operations on generic chainsof broken strings. Let ∂ denote the singular boundary operator, thus

∂sλ := s1 − s0, ∂S := S|∂∆2

for 1- resp. 2-chains. For the definition of string coproducts we need the following

Definition 5.4. — Let p(t) =∑mk=1 pkt

k and q(t) =∑mk=1 qkt

k, pk, qk ∈ R2, be realpolynomials with 〈p1, q1〉 < 0. A spike with ends (p, q) is a Cm-function f : [a, b]→ D2

with the following properties (see Figure 5.1):

J.É.P. — M., 2017, tome 4


U j

λjDδ

R2 Dε

×(−γ, γ) ×(−γ, γ)

×(−γ, γ)

ψ × 1l

∼=

σ f ∼= Φ× 1l

Φ Ψ

f

Figure 5.2. Construction of the map f .

(S1) the Taylor polynomials to order m of f at a resp. b agree with p resp. q;(S2) 〈f(t), p1〉 > 0 and 〈f(t), q1〉 < 0 for all t ∈ (a, b).

Remark 5.5. — Note that the spikes with fixed ends (p, q) and fixed or varying a < b

form a convex (hence contractible) space.

We choose a family of preferred spikes sp,q : [0, 1] → D2 for all (p, q) dependingsmoothly (with respect to the Cm-topology) on the coefficients of p and q. Now weare ready to define the string coproducts δN , δQ on generic d-chains for d 6 2.

d = 0. — On 0-chains set δN = δQ = 0.

d = 1. — For a 1-chain sλλ∈[0,1] let (λj , bj) be the finitely many values for whichsλ

j

2i (bj) ∈ K for some i = i(j). Set

δQsλ :=∑j

εj(sλ

j

1 , . . . , sλj

2i |[a2i−1,bj ], sj , sλ

j

2i |[bj ,a2i], . . . , sλj

2`+1

),

where sj = s(· − bj) : [bj , bj + 1] → N is a shift of the preferred spike s withends

(r∗T

mσλj

2i (bj), Tmσλ

j

2i (bj))in the normal directions, with constant value sλj2i (bj)

along K. The hat means shift by 1 in the argument, and εj = ±1 are the signs de-fined in Figure 2.2. Loosely speaking, δQ inserts an N -spike at all points where someQ-string meets K. The operation δN is defined analogously, inserting a Q-spike wherean N -string meets K. Note that by Definition 5.4 the spikes stay in N and meet Konly at their end points.

d = 2. — Finally, consider a generic 2-chain S : ∆2 → Σ`. Let λj ∈ int∆2 be thefinitely many points where σλji (aλ

j

i ) = 0 for some i = i(j). For the following construc-tion see Figure 5.2. Let δ > 0 be a number 6 1 such that the map ψ : λ 7→ σλi (aλi ) is adiffeomorphism from a neighborhood U j of λj onto the δ-disk Dδ ⊂ R2 (such δ existsby condition (2e) in Section 5.2). We choose U j so small that it contains no other λi.

J.É.P. — M., 2017, tome 4


Let γ > 0 be a number 6 1 such that |σλ(t + aλi )| 6 1 for all |t| 6 γ. Consider thefunction σ : U j × (−γ, γ)→ R2 defined by

σ(λ, t) :=

σλi (t+ aλi ) : t < 0,

−σλi+1(t+ aλi ) : t > 0 if i is even,σλi+1(t+ aλi ) : t > 0 if i is odd.

According to conditions (5.1), the function σ(λ, t) is smooth in λ and of class C2 butnot C3 in t. Define the function

f : Dδ × (−γ, γ) −→ R2, (a, b, t) 7−→ σ(ψ−1(a, b), t

).

By construction we have ∂f∂t (a, b, 0) = (a, b) for all (a, b). Moreover, by condition (2e)

in Section 5.2 we have vj := (σλj

i )(3)(aλj

i ) 6= 0. Let Ψ be the rotation of R2 whichmaps vj onto a vector (µ, 0) with µ > 0, let Φ : R2 → R2 be multiplication by 6/µ,and set ε := 6δ/µ. Then the map

(5.2) f := Φ Ψ f (Φ−1 × 1l) : Dε × (−γ, γ) −→ R2

satisfies

f(a, b, 0) = (0, 0),∂f

∂t(a, b, 0) = (a, b),

∂2f

∂t2(a, b, 0) = (0, 0),

∂f3

∂t3(0, 0, 0) = ±(6, 0)

for all (a, b). Here the map f is C2 but not C3, and the statement about the thirdderivative ∂f3/∂t3(0, 0, 0) means that it equals +(6, 0) from the left and −(6, 0) fromthe right. Therefore, f has a Taylor expansion (again considered for t 6 0 and t > 0

separately)

(5.3) f(a, b, t) =(at− sgn(t)t3, bt

)+O(|a||t|3 + |b||t|3 + |t|4).

Here to simplify notation we tacitly assume that the restrictions of f to t 6 0 andt > 0 are C4 rather than C3. The following argument carries over to the C3 case ifwe replace throughout O(|t|4) by o(|t|3).

Consider first the model case without higher order terms, i.e., the function

f0(a, b, t) =(at− sgn(t)t3, bt

).

Note that the first component at − sgn(t)t3 of f0 is exactly the function that weencountered at the end of Section 4.3. The zero set of f0 consists of three strata

t = 0 ∪ b = 0, a > 0, t =√a > 0 ∪ b = 0, a < 0, t = −

√−a < 0.

For a > 0 and b = 0 the function

fa : [0,√a] −→ R2, t 7−→ f0(a, 0, t) = (at− t3, 0)

is a spike with ends satisfying

f ′a(0) = (a, 0), f ′a(√a) = (−2a, 0), f ′′′a (0) = f ′′′a (

√a) = −6.

J.É.P. — M., 2017, tome 4


R× 0

−ε ε a

b

Figure 5.3. Two families of spikes vanishing at the origin.

Similarly, for a < 0 the function

fa : [−√−a, 0] −→ R2, t 7−→ f0(a, 0, t) = (at+ t3, 0)

is a spike with ends satisfying

f ′a(0) = (a, 0), f ′a(−√−a) = (−2a, 0), f ′′′a (0) = f ′′′a (−

√−a) = +6.

So two families of spikes pointing in the same directions come together from both sidesalong the a-axis b = 0 and vanish at (a, b) = (0, 0), see Figure 5.3. The followinglemma states that this qualitative picture persists in the presence of higher orderterms.

Lemma 5.6. — Let f : Dε × (−γ, γ) → R2 be a function satisfying (5.3). Thenfor ε and γ sufficiently small there exist smooth functions β(a, t) and τ(a) fora ∈ [−ε, ε]r0 such that with β(a) := β

(a, τ(a)

)the zero set of f in Dε × (−γ, γ)

consists of three strata

t = 0 ∪ b = β(a), a > 0, t = τ(a) > 0 ∪ b = β(a), a < 0, t = τ(a) < 0.

The functions β, τ satisfy the estimates

β(a, t) = O(|a|t2 + |t|3), τ(a)2 − a = O(a3/2), β(a) = O(a3/2).

Moreover, the functions

fa : [0, τ(a)] −→ R2, t 7−→ f(a, β(a), t

), a > 0,

fa : [τ(a), 0] −→ R2, t 7−→ f(a, β(a), t

), a < 0

are spikes with ends satisfying

f ′a(0) = (a, 0) +O(a3/2), f ′a(τ(a)

)= (−2a, 0) +O(a3/2).

Proof. — We consider the case a, t > 0, the case a, t < 0 being analogous. Settingthe second component in (5.3) to zero and dividing by t yields b = O(at2 + bt2 + t3),which for t sufficiently small can be solved for b = β(a, t) satisfying the estimate

J.É.P. — M., 2017, tome 4


β(a, t) = O(at2 + t3). Inserting this into the first component in (5.3), setting it to zeroand dividing by t yields

a− t2 = O(at2 + β(a, t)t2 + t3

)= O(at2 + t3),

which for (a, t) sufficiently small can be solved for t = τ(a) satisfying the estimateτ(a)2 − a = O(a3/2). Inserting t = τ(a) in β(a, t) we obtain the estimate β(a) =

O(a3/2). This proves the first assertions.Now consider the function fa(t) = f

(a, β(a), t

)for t ∈ [0, τ(a)] and a > 0. Inserting

β(a) = O(a3/2) we find

fa(t) = (at− t3, 0) +O(β(a)t+ at3 + β(a)t3 + t4

)= (at− t3, 0) +O(a3/2t+ at3 + t4),

and therefore f ′a(t) = (a−3t2, 0) +O(a3/2 +at2 + t3). This immediately gives f ′a(0) =

(a, 0) +O(a3/2) and, using τ(a) = O(a1/2), also f ′a(τ(a)

)= (−2a, 0) +O(a3/2).

It remains to prove that the functions fa are spikes in the sense of Definition 5.4.Write in components f = (f1, f2) and fa = (f1

a , f2a ) and abbreviate τ := τ(a). We

claim that there exist constants δ,D > 0 independent of a, t such that for all t ∈ [0, τ ]

we have

f1a (t) > 2δt(τ2 − t2), |f2

a (t)| 6 Dt(a+ t)(τ − t).

For the first estimate, note that

1

tf1a (t) = a− t2 +O(a3/2 + at2 + t3),

viewed as a function of t2, has transversely cut out zero locus t = τ and is therefore> 2δ(τ2 − t2) for some δ > 0. The second estimate holds because

1

tf2a (t) = O(a3/2 + at2 + t3)

vanishes at t = τ , so |f2a (t)| 6 Dt(a + t)(τ − t) for some constant D. Using these

estimates as well as f ′a(0) = (a, 0) + O(a3/2) and τ = O(a1/2) we compute with ageneric constant C (independent of a, t):

〈f ′a(0), fa(t)〉 =(a+O(a3/2)

)f1a (t) + 〈O(a3/2), f2

a (t)〉

> aδt(τ2 − t2)− Ca3/2t(τ − t)(a+ t)

= at(τ − t)(δ(τ + t)− Ca1/2(a+ t)

)> a3/2t(τ − t)

(δ − C(a+ t)

),

which is positive for 0 < t < τ and a sufficiently small. An analogous computation,using f ′a(τ) = (−2a, 0) +O(a3/2), shows 〈f ′a(τ), fa(t)〉 < 0, so fa is a spike.

J.É.P. — M., 2017, tome 4


Remark 5.7. — The spikes from Lemma 5.6 can be connected to the spike of themodel function f0 without higher order terms by rescaling: For s ∈ (0, 1] set

fs(a, b, t) :=1

s3f(s2a, s2b, st)

=(at− sgn(t)t3, bt

)+ sO(|a||t|3 + |b||t|3 + |t|4)

s→ 0−−−−−−→(at− sgn(t)t3, bt

).

Thus for |a| 6 ε the corresponding family of spikes (fsa)s∈[0,1] connects fa to thespike f0

a .

Now we return to the points λj ∈ U j and the corresponding maps

f : Dε × (−γ, γ) −→ R2

defined by (5.2). After shrinking ε, γ > 0 and replacing U j by (Φ ψ)−1(Dε) ⊂ ∆2

(where ψ,Φ are the maps defined above), we may assume that ε, γ satisfy the smallnessrequirement in Lemma 5.6 for each j. Define

MδQ:=⋃i

(ev2i S)−1(K) r⋃j

(U j × (0, 1)

),

MδN:=⋃i

(ev2i−1 S)−1(K) r⋃j

(U j × (0, 1)

).

By construction, MδQand MδN

are 1-dimensional submanifolds with boundary of∆2 × (0, 1). Define δQS : MδQ

→ Σ`+1 by inserting preferred N -spikes at all pointswhere some Q-string meets K (via the same formula as the one above for δQ on1-chains), and similarly for δNS. See Figure 5.4.

Note that the boundary ∂MδQconsists of intersections with ∂∆2 and with the

boundaries ∂U j . Thus each j contributes a unique point λjQ to ∂MδQ, which corre-

sponds in the above coordinates to a = +ε if the associated index i is odd and toa = −ε if i is even. Similarly, each j contributes a unique point λjN to ∂MδN

whichcorresponds in the above coordinates to a = −ε if the associated index i is odd andto a = +ε if i is even. The broken strings δQS(λjQ) and δNS(λjN ) are Cm-close for|t| > γ, and by Lemma 5.6 for |t| < γ they both have a Q-spike and an N -spike withthe same first derivatives at the ends. So, using convexity of the space of spikes withfixed ends (Remark 5.5, see also Remark 5.7), we can connect them by a short 1-chainSj : [0, 1]→ Σ`+1 with spikes in [−γ, γ] (which we regard as Q-spikes.)

We define δQS : MδQ → Σ`+1 to be δQS together with the 1-chains Sj , and weset δNS := δNS : MδN = MδN

→ Σ`+1. Recall that the 1-dimensional submanifoldMδQ

⊂ ∆2 × (0, 1) is the union of the transversely cut out preimages of K under theevaluation maps ev2i S : ∆2× (0, 1)→ Q. Hence the coorientation of K ⊂ Q and theorientation of ∆2 × (0, 1) induce an orientation on MδQ

, and similarly for MδN. (The

induced orientations depend on orientation conventions which will be fixed in theproof of Proposition 5.8 below.) We parametrize each connected component of MδQ

and MδNby the interval ∆1 = [0, 1] proportionally to arclength (with respect to

J.É.P. — M., 2017, tome 4


Sj

δNS

δQS

∆2

U j

λjN

λjQ

Figure 5.4. The definition of δQS = δQS + Sj and δNS = δNS.

the standard metric on ∆2 × (0, 1) and in the direction of the orientation, where forcomponents diffeomorphic to S1 we choose an arbitrary initial point). So we can viewδQS : MδQ → Σ`+1 and δNS : MδN → Σ`+1 as generic 1-chains, where we orient the1-chains Sj such that the points δQS(λjQ) appear with opposite signs in the boundaryof Sj and MδQ

.

Proposition 5.8. — On generic chains of degree 2, the operations ∂, δQ and δN satisfythe relations

∂2 = δ2Q = δ2

N = δQδN + δNδQ = 0,

∂δQ + δQ∂ + ∂δN + δN∂ = 0.

In particular, these relations imply

(∂ + δQ + δN )2 = 0.

Proof. — Consider a generic 2-chain S : ∆2 → Σ`. We continue to use the notationabove and denote by π : ∆2 × (0, 1) → ∆2 the projection. The relation ∂2S = 0

is clear. Points in δ2QS correspond to transverse self-intersections of π(MδQ

), so eachpoint appears twice with opposite signs, hence δ2

QS = 0 and similarly δ2NS = 0. Points

in δQδNS + δNδQS correspond to transverse intersections of π(MδQ) and π(MδN

),so again each point appears twice with opposite signs and the expression vanishes.Note that the broken strings corresponding to these points have two preferred spikesinserted at different places, so due to the uniqueness of preferred spikes with givenend points the broken strings do not depend on the order in which the spikes areinserted.

In order to achieve ∂δQ + δQ∂+ ∂δN + δN∂ = 0, we choose the orientation conven-tions for MδQ

and MδNsuch that (see Figure 5.4):

J.É.P. — M., 2017, tome 4


– ∂δQS+ δQ∂S corresponds to the intersection points λjQ of MδQwith the bound-

aries of the regions U j , and similarly for ∂δNS + δN∂S;– the sign of λjQ as a boundary point of MδQ

is opposite to the sign of λjN as aboundary point of MδN

.Due to the choice of the 1-chains Sj , it follows that ∂δQS + δQ∂S is the sum of thepoints δNS(λjN ) with suitable signs, and ∂δNS+δN∂S is the same sum with oppositesigns, so the total sum equals zero.

5.4. The string chain complex. — For d = 0, 1, 2 and ` > 0 let Cd(Σ`) be the freeZ-module generated by generic d-chains in Σ`, and set

Cd(Σ) :=∞⊕=0

Cd(Σ`), d = 0, 1, 2.

The string operations defined in Section 5.3 yield Z-linear maps

∂ : Cd(Σ`) −→ Cd−1(Σ`), δN , δQ : Cd(Σ

`) −→ Cd−1(Σ`+1).

The induced maps ∂, δQ, δN : Cd(Σ) → Cd−1(Σ) satisfy the relations in Proposi-tion 5.8, in particular

D := ∂ + δQ + δN .

satisfies D2 = 0. We call(C∗(Σ), ∂+ δQ + δN

)the string chain complex of K, and we

define the degree d string homology of K as the homology of the resulting complex,

Hstringd (K) := Hd

(C∗(Σ), ∂ + δQ + δN

), d = 0, 1, 2.

Concatenation of broken strings at the base point x0 (and the canonical subdivisionof ∆1 ×∆1 into two 2-simplices) yields products

× : Cd(Σ`)× Cd′(Σ`

′) −→ Cd+d′(Σ

`+`′), d+ d′ 6 2

satisfying the relations

(5.4) (a× b)× c = a× (b× c), D(a× b) = Da× b+ (−1)deg aa×Db

whenever deg a+ deg b+ deg c 6 2. In particular, this gives C0(Σ) the structure of a(noncommutative but strictly associative) algebra over Z and C1(Σ), C2(Σ) the struc-ture of bimodules over this algebra. These structures induce on homology the structureof a Z-algebra on Hstring

0 (K), and of bimodules over this algebra on Hstring1 (K) and

Hstring2 (K). By definition, the isomorphism classes of the algebra Hstring

0 (K) and themodules Hstring

1 (K), Hstring2 (K) are clearly isotopy invariants of the framed oriented

knot K.We can combine these invariants into a single graded algebra as follows. For d > 2,

we define Cd(Σ`) to be the free Z-module generated by products S1×· · ·×Sr of genericchains Si of degrees 1 6 di 6 2 in Σ`i such that d1 + · · ·+dr = d and `1 + · · ·+ `r = `,modulo the submodule generated by

S1 × · · · × Sr − S′1 × · · · × S′r′

J.É.P. — M., 2017, tome 4


for different decompositions of the same d-chain. Put differently, this submodule isgenerated by

S1 × · · ·Si × Si+1 × · · · × Sr − S1 × · · · (Si × Si+1)× · · · × Sr,

where Si and Si+1 are generic 1-chains and (Si × Si+1) is the associated generic 2-chain. Note that for d = 2 this definition of C2(Σ`) agrees with the earlier one. Wedefine D = ∂ + δQ + δN on

Cd(Σ) :=∞⊕=0

Cd(Σ`), d > 0

by the Leibniz rule. This is well-defined in view of the second equation in (5.4) andsatisfies D2 = 0. Together with the product × this gives C∗(Σ) the structure of adifferential graded Z-algebra. The total string homology

Hstring∗ (K) := H∗

(C∗(Σ), D

)inherits the structure of a graded Z-algebra whose isomorphism class is an invariantof the framed oriented knot K.

Remark 5.9. — Our definition of string homology of K in degrees > 2 in terms ofproduct chains is motivated by Legendrian contact homology of ΛK when Q = R3

which is then generated by elements of degrees 6 2. From the point of view of stringtopology, it would appear more natural to define string homology in arbitrary degreesin terms of higher dimensional generic chains of broken strings in the sense of Defi-nition 5.3. Similarly, for knot contact homology in other ambient manifolds, e.g. forQ = S3, there are higher degree Reeb chords that contribute to the (linearized) con-tact homology. It would be interesting to see whether such constructions would carryadditional information.

5.5. Length filtration. — Up to this point, the constructions have been fairly sym-metric in the Q-and N -strings. However, as we will see below, the relation to Leg-endrian contact homology leads us to assign to Q-strings s2i their geometric lengthL(s2i), and to N -strings length zero. Thus we define the length of a broken strings = (s1, . . . , s2`+1) by

L(s) :=∑i=1

L(s2i),

where we do not include in the sum those s2i that are Q-spikes in the sense of Defi-nition 5.4. We define the length of a generic i-chain S : K → Σ by

L(S) := maxk∈K

L(S(k)

).

Then the subspaces

F `Ci(Σ) :=∑

ajSj ∈ Ci(Σ) | L(Sj) 6 ` whenever aj 6= 0

define a filtration in the sense that F kCi(Σ) ⊂ F `Ci(Σ) for k 6 ` and

D(F `Ci(Σ)

)⊂ F `Ci−1(Σ).

J.É.P. — M., 2017, tome 4


This length filtration will play an important role in the proof of the isomorphism toLegendrian contact homology in Section 7.

Remark 5.10. — The omission of the length of Q-spikes from the length of a brokenstring ensures that the operation δN , which inserts Q-spikes, does not increase thelength. Since Q-spikes do not intersect the knot in their interior, they are not affectedby δQ and it follows that D preserves the length filtration.

6. The chain map from Legendrian contact homology to string homology

In this section we define a chain map Φ: C∗(R)→ C∗(Σ) from a complex comput-ing Legendrian contact homology to the string chain complex defined in the previoussection. The boundary operator on C∗(R) is defined using moduli spaces of holomor-phic disks in R×S∗Q with Lagrangian boundary condition R×ΛK and the map Φ isdefined using moduli spaces of holomorphic disks in T ∗Q with Lagrangian boundarycondition Q ∪ LK , where the boundary is allowed to switch back and forth betweenthe two irreducible components of the Lagrangian at corners as in Lagrangian inter-section Floer homology. We will describe these spaces and their properties, as wellas define the algebra and the chain map. In order not to obscure the main lines ofargument, we postpone the technicalities involved in detailed proofs to Sections 8–10.

6.1. Holomorphic disks in the symplectization. — Consider a contact (2n − 1)-manifold (M,λ) with a closed Legendrian (n− 1)-submanifold Λ. For the purposes ofthis paper we only consider the case that M = S∗Q is the cosphere bundle of Q = R3

with its standard contact form λ = p dq and Λ = ΛK is the unit conormal bundleof an oriented framed knot K ⊂ Q, but the construction works more generally forany pair (M,Λ) for which M has no contractible closed Reeb orbits, see Remark 6.4below.

Denote by R the Reeb vector field of λ. A Reeb chord is a solution a : [0, T ]→ M

of a = R with a(0), a(T ) ∈ Λ. Reeb chords correspond bijectively to binormal chordsof K, i.e., geodesic segments meeting K orthogonally at their endpoints. As usual,we assume throughout that Λ is chord generic, i.e., each Reeb chord corresponds toa Morse critical point of the distance function on K ×K.

In order to define Maslov indices, one usually chooses for each Reeb chorda : [0, T ] → M capping paths connecting a(0) and a(T ) in Λ to a base point x0 ∈ Λ.Then one can assign to each a completed by the capping paths a Maslov index µ(a),see [5, App.A]. In the case under consideration (M = S∗R3 and Λ = ΛK) the Maslovclass of Λ equals 0, so the Maslov index does not depend on the choice of cappingpaths. It is given by µ(a) = ind(a)+1, where ind(a) equals the index of a as a criticalpoint of the distance function on K × K, see [12]. We define the degree of a Reebchord a as

|a| := µ(a)− 1 = ind(a),

J.É.P. — M., 2017, tome 4


and the degree of a word b = b1b2 · · · bm of Reeb chords as

|b| :=m∑j=1

|bj |.

Given a and b, we write M sy(a; b) for the moduli space of J-holomorphic disksu : (D, ∂D)→ (R×M,R×Λ) with one positive boundary puncture asymptotic to theReeb chord strip over a at the positive end of the symplectization, and m negativeboundary punctures asymptotic to the Reeb chord strips over b1, . . . , bm at the neg-ative end of the symplectization. Here J is an R-invariant almost complex structureon R×M compatible with λ. For generic J , the moduli space M sy(a; b) is a manifoldof dimension

dim(M sy(a; b)) = |a| − |b| = |a| −m∑j=1

|bj |,

see Theorem 10.1. In fact, the moduli spaces correspond to the zero set of a Fredholmsection of a Banach bundle that can be made transverse by perturbing the almostcomplex structure, and there exist a system of coherent (or gluing compatible) ori-entations of the corresponding index bundles over the configuration spaces and thissystem induces orientations on all the moduli spaces.

By our choice of almost complex structure, R acts on M sy(a; b) by translationsin the target R ×M and we write M sy(a; b)/R for the quotient, which is then anoriented manifold of dimension |a| − |b| − 1.

Finally, we discuss the compactness properties of M sy(a; b)/R. The moduli spaceM sy(a; b)/R is generally not compact but admits a compactification by multileveldisks, where a multilevel disk is a tree of disks with a top level disk in M sy(a, b1),b1 = b11 · · · b1m1

, second level disks in M sy(b1i ; b2,i) attached at the negative punctures

of the top level disk, etc. See Figure 6.9 below. It follows from the dimension formulaabove that the formal dimension of the total disk that is the union of the levelsin a multilevel disk is the sum of dimensions of all its components. Consequently,for generic almost complex structure, if dim(M sy(a; b)) = 1 then M sy(a; b)/R isa compact 0-dimensional manifold, and if dim(M sy(a; b)) = 2 then the boundaryof M sy(a; b)/R consists of two-level disks where each level is a disk of dimension 1

(and possibly trivial Reeb chord strips).The simplest version of Legendrian contact homology would be defined by the free

Z-algebra generated by the Reeb chords, with differential counting rigid holomorphicdisks. In the following subsection we will define a refined version which also incorpo-rates the boundary information of holomorphic disks.

6.2. Legendrian contact homology. — In this subsection we define a version ofLegendrian contact homology that will be directly related to the string homologyof Section 5, see [10] for a similar construction in rational symplectic field theory. Theusual definition of Legendrian contact homology is a quotient of our version. We keepthe notation from Section 6.1.

J.É.P. — M., 2017, tome 4


a1 a2 = a a3

b1 b2 b3

α1

α2α3

α4

β1

β2 β3

β4

x0

u

Figure 6.1. The definition of ∂(u) and ∂(u) ·i a.

Fix an integer m > 3. For points x, y ∈ Λ we denote by Px,yΛ the space of Cmpaths γ : [a, b]→ Λ with γ(a) = x and γ(b) = y whose firstm derivatives vanish at theendpoints. Here the interval [a, b] is allowed to vary. The condition at the endpointsensures that concatenation of such paths yields again Cm paths. Fix a base pointx0 ∈ Λ and denote by Ωx0

Λ = Px0x0Λ the Moore loop space based at x0.

Definition 6.1. — A Reeb string with ` chords is an expression α1a1α2a2 · · ·α`a`α`+1,where the ai : [0, Ti]→M are Reeb chords and the αi are elements in the path spaces

α1 ∈ Px0a1(T1), αi ∈ Pai−1(0)ai(Ti) for 2 6 i 6 `, α`+1 ∈ Pa`(0)x0.

See the top of Figure 6.1. Note that the αi and the negatively traversed Reebchords ai fit together to define a loop in M starting and ending at x0. Concatenatingall the αi and ai in a Reeb string with the appropriate capping paths, we can vieweach αi as an element in the based loop space Ωx0

Λ. However, we will usually nottake this point of view.

Boundaries of holomorphic disks in the symplectization give rise to Reeb stringsas follows. Consider a holomorphic disk u belonging to a moduli space M sy(a; b)

as above, with Reeb chords a : [0, T ] → M and bi : [0, Ti] → M , i = 1, . . . , `. Itsboundary arcs in counterclockwise order and orientation projected to Λ define pathsβ1, . . . , β` in Λ as shown in Figure 6.1, i.e.,

β1 ∈ Pa(T )b1(T1), βi ∈ Pbi−1(0)bi(Ti) for 2 6 i 6 `, β`+1 ∈ Pb`(0)a(0).

J.É.P. — M., 2017, tome 4


We denote the alternating word of paths and Reeb chords obtained in this way as theboundary of u by

(6.1) ∂(u) := β1b1β2b2 · · ·β`b`β`+1.

Note that the βi and the negatively traversed Reeb chords bi fit together to definea path in M from a(T ) to a(0). We obtain from ∂(u) a Reeb string if we extend β1

and β`+1 to the base point x0 by the capping paths of a.For ` > 0 we denote by R` the space of Reeb strings with ` chords, equipped with

the Cm topology on the path spaces. Note that different collections of Reeb chordscorrespond to different components. Concatenation at the base point gives

R := q`>0R`

the structure of an H-space. Note that the sub-H-space R0 = Ωx0Λ agrees with theMoore based loop space with its Pontrjagin product. Let

C(R) =⊕d>0

Cd(R)

be singular chains in R with integer coefficients. It carries two gradings: the degree das a singular chain, which we will refer to as the chain degree, and the degree

∑`i=1 |bi|

of the Reeb chords, which we will refer to as the chord degree. For sign rules we thinkof the chain coming first and the Reeb chords last. The total grading is given by thesum of the two degrees. Recall that it does not depend on the choice of capping paths.Concatenation of Reeb strings at the base point and product of chains gives C(R)

the structure of a (noncommutative but strictly associative) graded ring. Note thatit contains the subring

C(R0) = C(Ωx0Λ).

Next we define the differential

∂Λ = ∂sing + ∂sy : C(R) −→ C(R).

Here ∂sing is the singular boundary and ∂sy is defined as follows. Pick a genericcompatible cylindrical almost complex structure J on the symplectization R ×M .Consider a punctured J-holomorphic disk u : D → R ×M in M sy(a; b). If the Reebchord a = ai appears in a Reeb string a = α1a1 . . . amαm+1, then we can replace aiby ∂(u) to obtain a new Reeb string which we denote by

∂(u) ·i a := α1a1 · · · αi∂(u)αi+1 · · · a`α`+1.

Here ∂(u) is defined in (6.1) and the paths αi, αi+1 are the concatenations of αi, αi+1

with the paths β1, β`+1 in ∂(u), respectively. See Figure 6.1. For a chain a ∈ C(R) ofReeb strings of type a = α1a1 · · · amαm+1 we now define

∂sy(a) :=∑i=1

∑|ai|−|b|=1

u∈M sy(ai;b)/R

ε(−1)d+|a1|+···+|ai−1|∂(u) ·i a,

where d is the chain degree of a and ε is the sign from the orientation of M sy(ai; b)/Ras a compact oriented 0-manifold (i.e., points with signs). Note that ∂sy preserves the

J.É.P. — M., 2017, tome 4


chain degree and decreases the chord degree by 1, whereas ∂sing preserves the chorddegree and decreases the chain degree by 1. In particular, ∂Λ has degree −1 withrespect to the total grading. The main result about the contact homology algebrathat we need is summarized in the following theorem.

Theorem 6.2. — The differential ∂Λ : C(R)→ C(R) satisfies ∂2Λ = 0 and the Legen-

drian contact homology

Hcontact(Λ) := ker ∂Λ/ im ∂Λ

is independent of all choices.

Proof. — In the case that we use it, for M = S∗R3 and Λ = ΛK , the proof is an easyadaptation of the one in [15, 14] and [7], see also [12].

Consider first the equation for the differential. The equation ∂2Λ = 0 follows from

our description of the boundary of the moduli spaces M sy(a; b) of dimension 2 inSection 6.1, which shows that contributions to (∂sy)2 are in oriented one-to-one cor-respondence with the boundary of an oriented 1-manifold and hence cancel out. Therelations (∂sing)2 = 0 and ∂sing∂sy + ∂sy∂sing = 0 are clear.

To prove the invariance statement we use a bifurcation method similar to [14, §4.3].Consider a generic 1-parameter family (Λs, Js), s ∈ S = [0, 1], of Legendrian subman-ifolds and almost complex structures. By genericity of the family there is a finite setof points s1 < s2 < · · · < sm such that in S r s1, . . . , sm all Reeb chords of Λsare transverse, all Reeb chords have distinct actions, and all holomorphic disks deter-mined by (Λs, Js) have dimension at least 1 (i.e., if we write M sy

s for moduli spacesdetermined by (Λs, Js) then dim M sy

s (a, b) > 1 if the moduli space is nonempty).Furthermore, the points sj are of three kinds:

– handle slides, where all Reeb chords are nondegenerate but where there is atransversely cut out disk of formal dimension 0 (i.e., there exists one M sy(a; b), withdim M sy(a; b) = 0 which contains one R-family of Jsj -holomorphic disks with bound-ary on Λsj , and this disk is transversely cut out as a solution of the parameterizedproblem);

– action switches, where two nondegenerate Reeb chords have the same action andtheir actions interchange;

– birth/death moments where there is one degenerate Reeb chord at which twoReeb chords cancel through a quadratic tangency.

To show invariance we first observe that if [s′, s′′] ⊂ S is an interval which does notcontain any sj , then the Reeb chords of Λs, s ∈ [s′, s′′] form 1-manifolds canonicallyidentified with [s′, s′′] and the actions of the different Reeb chord manifolds do notcross. Thus for Reeb chords a, b1, . . . , bm of Λs′ we get corresponding chords on Λsfor each s ∈ [s′, s′′] = S′ which we denote by the same symbols, suppressing thes-dependence below. We next define a chain map

Φ: C(Rs′) −→ C(Rs′′)

J.É.P. — M., 2017, tome 4


which counts geometrically induced chains as follows. We introduce the notion of adisk with lines of Reeb chords. Such an object has a positive puncture at the Reebchord a over s′ and negative punctures at Reeb chords according to b over s′′ andis given by a collection of disks u1, . . . , um where the disk uj is a disk at σj , wheres′ 6 σ1 6 σ2 6 · · · 6 σm 6 s′′ and if σj > σk for some k then its positive Reeb chord isconnected by a line in a Reeb chord manifold to a Reeb chord at the negative punctureof some ur for σr < σj . The collection of such objects naturally forms a moduli space,M sy

S′ (a; b), where we glue two disks when the length of the line connecting them goesto zero. We define the chain map Φ as

Φ(a) = [M syS′ (a, b)] .

The chain map equation ∂Λs′′Φ = Φ∂Λs′ follows immediately once one notices that thecodimension one boundary of the moduli space consists of disks over the endpointswith lines of Reeb chords over [s′, s′′] attached. (We point out that this constructionis inspired by Morse-Bott arguments, compare [16].)

Consider the filtration in C(R) which associates to a chain of Reeb strings thesum of actions of its Reeb chords. By Stokes’ theorem the differential respects thefiltration. The pure lines of Reeb chords (without disks) contribute to the map andshow that

Φ(a) = a+ Φ0(a),

where the action of Φ0(a) is strictly smaller than that of a. It follows that Φ inducesan isomorphism on the E2-page of the action spectral sequence and hence is a quasi-isomorphism.

In order to show invariance at the bifurcation moments we consider the deformationin a small interval around [sj − ε, sj + ε]. In this case we can construct a Lagrangiancobordism L in the symplectization R ×M interpolating between the cylinders onΛsj−ε and Λsj+ε, see [10, Lem.A.2]. If a is a Reeb chord of Λsj+ε and b is a word ofReeb chords of Λsj−ε then let M sy,L(a; b) denote the moduli space of holomorphicdisks defined as M sy(a, b), see Section 6.1, but with boundary condition given by Linstead of R × Λ. (Note that since L is not R-invariant, R does generally not act onM sy,L(a; b).) We define a chain map

Φ: C(R+) −→ C(R−)

between the algebras at the positive and the negative ends as follows: Φ is the identitymap on chains, and on Reeb chords a of Λsj−ε Φ is given by

Φ(a) =∑b

[M sy,L(a; b)],

where b runs over all words of Reeb chords of Λsj+ε and [M sy,L(a; b)] denotes thechain of Reeb strings carried by the moduli space. SFT compactness and gluing asin [10] shows that the chain map equation ∂Λsj+ε

Φ = Φ∂Λsj−εholds. It remains to

show that Φ is a quasi-isomorphism.

J.É.P. — M., 2017, tome 4


Consider first the case that sj is a handle slide. Taking ε sufficiently small we findthat for each Reeb chord a on Λsj−ε there is a unique holomorphic strip connecting itto the corresponding Reeb chord a on Λsj+ε. (These strips converge to trivial stripsas ε→ 0.) It follows that for each generator c (chord or chain),

Φ(c) = c+ Φ0(c),

where the filtration degree of Φ0(c) is strictly smaller than that of c. Thus Φ inducesan isomorphism on the E2-page of the action filtration spectral sequence and is hencea quasi-isomorphism.

Consider second the case of an action switch. In this case we find exactly as in thehandle slide case that

Φ(c) = c+ Φ0(c)

for each c. The only difference is that now one action window contains two generators.Since the two Reeb chords have the same action but lie at a positive distance apart, itfollows by monotonicity and Stokes’ theorem that the chain map induces an isomor-phism also in this action window. We find as above that Φ is a quasi-isomorphism.

Finally consider the case that sj is a birth moment where two new Reeb chords aand b are born (the death case is analogous). For ε > 0 sufficiently small we have

∂sya = b+ ∂sy0 (a),

where the action of ∂sy0 (a) is strictly smaller than the action of b, see [13, Lem. 2.14].

As above we find that for any Reeb chord c of Λsj+ε we have

Φ(c) = c+ Φ0(c).

If we filter by small action windows that contain one Reeb chord each, except forone that contains both a and b (note that the action of a approaches the action of bas ε → 0) we find again that Φ gives an isomorphism on the E2-page and hence isan isomorphism. We conclude that we can subdivide the interval S into pieces withendpoints with quasi-isomorphic algebras. The theorem follows.

According to Theorem 6.2, (C(R), ∂Λ) is a (noncommutative but strictly associa-tive) differential graded (dg) ring containing the dg subring(

C(R0), ∂Λ

)=(C(Ωx0

Λ), ∂sing).

Thus (C(R), ∂Λ) is a (C(R0), ∂Λ)-NC-algebra in the sense of the following definition.

Definition 6.3. — Let (R, ∂) be a dg ring. An (R, ∂)-NC-algebra is a dg ring (S, ∂S)

together with a dg ring homomorphism (R, ∂)→ (S, ∂S).

It follows that the Legendrian contact homology Hcontact(Λ) is an NC-algebra overthe graded ring

H∗(Ωx0Λ, ∂sing) ∼= Zπ1(Λ) ∼= Z[λ±1, µ±1].

Here we have used that in our situation Λ ∼= T 2 is a K(π, 1), so all the homology ofits based loop space is concentrated in degree zero and agrees with the group ring ofits fundamental group π1(Λ) ∼= Z2.

J.É.P. — M., 2017, tome 4


Relation to standard Legendrian contact homology. — Recall that C(R) is a doublecomplex with bidegree (chain degree, chord degree), horizontal differential ∂sing, andvertical differential ∂sy. As observed above, the first page of the spectral sequencecorresponding to the chord degree is concentrated in the 0-th column and given by(

A := H0(R, ∂sing), ∂sy).

Generators of A are words α1a1α2a2 · · ·α`a`α`+1 consisting of Reeb chords ai andhomotopy classes of paths αi satisfying the same boundary conditions as before. Notethat A is an NC-algebra over the subring A 0 = H0(R0) ∼= Zπ1(Λ) (on which ∂Λ

vanishes), and A k = H0(Rk) is the k-fold tensor product of the bimodule A 1 overthe ring A 0.

We denote byA := A /I

the quotient of A by the ideal I generated by the commutators [a, β] of Reebchords a and β ∈ π1(Λ). Since ∂Λ(I ) ⊂ I , the differential descends to a differ-ential ∂sy

: A → A whose homology

Hcontact

(Λ) := ker ∂sy/ im ∂

sy

is the usual Legendrian contact homology as defined in [13].

Length filtration. — The complex (C(R), ∂Λ) is filtered by the length

L(α1a1α2a2 · · ·α`a`α`+1) :=∑i=1

L(ai),

where L(a) =∫aλ denotes the action of a Reeb chord a, which agrees with its period

and also with the length of the corresponding binormal cord. The length is preservedby the singular boundary operator ∂sing and strictly decreases under ∂sy.

Remark 6.4. — The construction of Legendrian contact homology in this subsectionworks for any pair (M,Λ) such that M has no contractible closed Reeb orbits. Ex-amples include cosphere bundles S∗Q of n-manifolds Q with a metric of nonpositivecurvature that are convex at infinity, with Λ = ΛK the unit conormal bundle of aclosed connected submanifold K ⊂ Q. However, if Λ is not a K(π, 1), then the co-efficient ring H∗(Ωx0

Λ, ∂sing) will not be equal to the group ring of its fundamentalgroup but contain homology in higher degrees.

6.3. Switching boundary conditions, winding numbers, and length. — We continueto consider Q = R3 equipped with the flat metric and an oriented framed knotK ⊂ Q. In addition, we assume from now on that K is real analytic; this can alwaysbe achieved by a small perturbation of K not changing its knot type. We equip T ∗Qwith an almost complex structure J which agrees with an R-invariant almost complexstructure on the symplectization of S∗Q outside a finite radius disk sub-bundle of T ∗Qand with the standard almost complex structure Jst on T ∗Q inside the disk sub-bundleof half that radius. An explicit formula for such J is given in Section 8.2. We point

J.É.P. — M., 2017, tome 4


1

−1 1

Figure 6.2. The biholomorphism χ.

out that the canonical isomorphism (T ∗Q, Jst) ∼= (C3, i) identifies the fibre with R3

and the zero section with iR3. Recall that L = Q ∪ LK .Let D be the closed unit disk with a boundary puncture at 1 ∈ ∂D and let

u : (D, ∂D)→ (T ∗Q,L) be a holomorphic disk with one positive puncture and switch-ing boundary conditions. This means that the map u is asymptotic to a Reeb chordat infinity at the positive puncture 1 and that it is smooth outside an additionalfinite number of boundary punctures where the boundary switches, i.e., jumps fromone irreducible component of L to another (which may be the same one). At theseadditional boundary punctures, the holomorphic disk is asymptotic to some pointin the clean intersection K ⊂ L, i.e., it looks like a corner of a disk in Lagrangianintersection Floer homology.

The real analyticity of K allows us to get explicit local forms for holomorphic disksnear corners. We show in Lemma 8.6 that there are holomorphic coordinates

R× (0, 0) ⊂ U ⊂ C× C2,

in which K corresponds to R× (0, 0), the 0-section Q corresponds to R×R2, and theconormal LK to R× iR2.

Consider now a neighborhood of a switching point of a holomorphic disk u on theboundary of D, where we use z in a half-disk D+

ε around 0 in the upper half-plane asa local coordinate around the switching point in the source. According to Section 4.1,u admits a Taylor expansion around 0, with u = (u1, u2) ∈ C× C2:

(6.2) u1(z) =∑k∈N

bkzk, u2(z) =

∑k∈ 1

2N

ckzk.

Here compared to Section 4.1 we have divided the indices by 2, so the bk and ckcorrespond to the a2k in Section 4.1. The coefficients bj are real constants, reflectingsmoothness of the tangent component of u. The ck satisfy one of the conditions inRemark 4.1, i.e., they are either all real or all purely imaginary vectors in C2, andthe indices are either all integers or all half-integers.

Equivalently (and more adapted to the analytical study in Sections 8 – 10) one canuse z in a neighborhood of infinity in the strip R× [0, 1] as a local coordinate in thesource. Composing the Taylor expansions (6.2) with the biholomorphism

(6.3) χ : R>0 × [0, 1]∼=−→ D+, z 7−→ − exp(−πz)

(see Figure 6.2) one gets instead the Fourier expansions

J.É.P. — M., 2017, tome 4


(6.4) u1(z) =∑k∈N

(−1)kbke−kπz, u2(z) =

∑k∈ 1

2N

(−1)kcke−kπz.

Recall from Section 4.2 that the local winding number at the switch is the positivehalf-integer or integer which is the index of the first non-vanishing Fourier coefficientsin the expansion of u2 in (6.4). The sum of the local winding numbers at all switchingboundary punctures is the total winding number of the disk. Since the number ofswitches from LK to Q equals that from Q to LK , the total winding number is aninteger.

The following technical result, which is a special case of [5, Th. 1.2], will play acrucial role in the sequel.

Theorem 6.5 ([5]). — For a cord-generic real analytic knot K ⊂ R3 the total wind-ing number, and in particular the number of switches, of any holomorphic disku : (D, ∂D) → (T ∗Q,L) with one positive puncture is uniformly bounded by aconstant κ.

Remark 6.6. — The necessary energy bound appearing in the corresponding state-ment in [5] is automatic here, since in our present situation the energy is given by theaction of the Reeb chord at the positive puncture, which only varies in a finite set.

In view of this result, when we discuss compactness we need only consider sequencesof holomorphic disks with a fixed finite number of switches, each of fixed windingnumber. As we prove in Section 8, each moduli space of such holomorphic disks is forgeneric data a manifold that admits a natural compactification as a manifold withboundary with corners. We will specifically need such moduli spaces of dimension 0, 1,or 2 and we give brief descriptions in these cases.

Let a be a Reeb chord of ΛK . Let q1, . . . , qm be punctures in ∂D and let n =

(n1, . . . , nm) be a vector of local winding numbers, so nj ∈

12 , 1,

32 , 2, . . .

is the local

winding number at qj . We write M (a;n) for the moduli space of holomorphic diskswith positive puncture at the Reeb chord a and switching punctures at q1, . . . , qmwith winding numbers according to n. Define the nonnegative integer

|n| :=m∑j=1

2(nj − 12 ) > 0.

Theorem 6.7. — For generic almost complex structure J , the moduli space M (a;n)

is a manifold of dimension

dim M (a;n) = |a| − |n|.

Furthermore, the choice of a spin structure on LK together with the spin structureon R3 induces a natural orientation on M (a;n).

Proof. — This is a consequence of [5, Th.A.1] and Lemma 9.5 below.

Note that, due to Theorem 6.5, any moduli space M (a;n) is empty if n has morethan κ components, i.e., there are more than κ switches.

J.É.P. — M., 2017, tome 4


12

12

12

12

12

12

12 1 1

212

12

N N N N

N N N

Q Q Q Q Q

a a

Figure 6.3. Type (Lag) boundary where an N -string disappears.

6.4. Moduli spaces of dimension zero and one. — For moduli spaces of dimension6 1 with positive puncture at a Reeb chord of degree 6 1, we have the following.Theorem 6.7 implies that if |a| = 0 then M (a;n) is empty if |n| > 0 and is otherwisea compact oriented 0-manifold. Likewise, if |a| = 1 then M (a;n) is empty if |n| > 1

and is an oriented 0-manifold if |n| = 1. Note that |n| = 1 implies that there isexactly one switch with winding number 1 and that the winding numbers at all otherswitches equal 1

2 . Finally, if all entries in n equal 12 then dim(M (a;n)) = 1.

It follows by Theorem 10.3 that the 1-dimensional moduli spaces of disks withswitching boundary condition admit natural compactifications to 1-manifolds withboundary. The next result describes the disk configurations corresponding to theboundary of these compact intervals.

Proposition 6.8. — If a is a Reeb chord of degree |a| = 1 and if all entries of n

equal 12 , then the oriented boundary of M (a;n) consists of the following:

(Lag) Moduli spaces M (a;n′), where n′ is obtained from n by removing two con-secutive 1

2 -entries and inserting in their place a 1.(sy) Products of moduli spaces

M sy(a; b)/R × Πbj∈b M (bj ;nj),

where n equals the concatenation of the nj.

Proof. — This is a consequence of Theorem 10.3. To motivate the result, note that thefirst type of boundary corresponds to two switches colliding, see Figures 6.3 and 6.4.The second type corresponds to a splitting into a two level curve with one R-invariantlevel (of dimension 1) in the symplectization and one rigid curve (of dimension 0)in T ∗Q, see Figure 6.5. By transversality, compactness, and the dimension formulathis accounts for all the possible boundary phenomena, and by a gluing argument wefind that any such configuration corresponds to a unique boundary point.

We conclude this subsection by giving an alternate interpretation of the first bound-ary phenomenon in Proposition 6.8. Let M ∗(a;n) denote the moduli space corre-sponding to M (a;n), but with one extra marked point on the boundary of the disk.

J.É.P. — M., 2017, tome 4


12

12

12

12

12

12

12

12

12

12

1

N N N N

N N N

Q Q Q Q Q

a a

Figure 6.4. Type (Lag) boundary where a Q-string disappears.

12

12

12

12

12

12

12

12

12

12

12

12

N

N N

N

N NN NN

R× ΛK R× ΛKR× ΛK

Q Q Q

Q Q Q

a

a

b1 b2

Figure 6.5. Type (sy) boundary.

Then M ∗(a;n) fibers over M (a;n) with fiber ∂D − 1, q1, . . . , qm and there is anevaluation map ev : M ∗(a;n) → L. It follows from Theorem 10.8 that for |a| = 1

and |n| = 0 (and generic data), ev−1(K) is a transversely cut out oriented 0-manifoldthat projects injectively into M (a;n). We denote its image by

δM (a;n).

As the notation suggests, this space will be the natural domain for the string opera-tions δ = δQ + δN .

Proposition 6.9. — If a is a Reeb chord of degree |a| = 1 and if all entries n equal 12 ,

then there is a natural orientation preserving identification between δM (a;n) andM (a;n′′), where n′′ is obtained from n by inserting in n a new entry equal to 1 atthe position given by the marked point.

Proof. — This is a consequence of Theorem 10.8. Here is the idea. Consider localcoordinates around the marked point in the source and around K in the target. Thenthe Taylor expansions (6.2) with c 1

2= 0 and c1 6= 0 give the map in δM (a;n) with the

marked point corresponding to 0. The corresponding Fourier expansions (6.4) present

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1

1 1

1

Figure 6.6. Type (Lag|Lag)1 corner.

the map as an element in M (a;n′′), where the marked point is replaced by a puncture.Conversely, translating the Fourier picture to the Taylor picture proves the otherinclusion and hence equality holds. See Section 9.6 for a discussion of orientations ofthe moduli spaces involved.

6.5. Moduli spaces of dimension two. — For moduli spaces M (a;n) with positivepuncture at a Reeb chord a of degree |a| = 2, Theorem 6.7 implies the following:

– If |n| > 2 then M (a;n) = ∅.– If |n| = 2 then M (a;n) is a compact 0-dimensional manifold. This can happen

in two ways: either exactly one entry in n equals 32 , or exactly two entries equal 1

and all others equal 12 .

– If |n| = 1 then M (a;n) is an oriented 1-manifold, exactly one entry in n equals 1

and all others equal 12 .

– If |n| = 0 then M (a;n) is an oriented 2-manifold and all entries in n equal 12 .

It follows by Theorem 10.6 that the 2-dimensional moduli spaces of disks withswitching boundary condition admit natural compactifications to 2-manifolds withboundary and corners. The next result describes the disk configurations correspondingto the boundary and corner points of these compact surfaces, see Figures 6.6, 6.7, 6.8and 6.9.

Proposition 6.10. — If a is a Reeb chord of degree |a| = 2 and if all entries of nequal 1

2 , then the 1-dimensional boundary segments in the boundary of M (a;n) consistof the following configurations:

(Lag) Moduli spaces M (a;n′), where n′ is obtained from n by removing two con-secutive 1

2 -entries and inserting in their place a 1.

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1

1

3/2

Figure 6.7. Type (Lag|Lag)2 corner.

11

Figure 6.8. Type (sy|Lag) corner.

(sy) Products of moduli spaces


where n equals the concatenation of the nj.The corner points in the boundary consists of the following configurations:

J.É.P. — M., 2017, tome 4


T

Figure 6.9. Type (sy|sy) corner.

(Lag|Lag)1 Moduli spaces M (a;n′), where n′ is obtained from n by removing twopairs of consecutive 1

2 -entries and inserting 1’s in their places.(Lag|Lag)2 Moduli spaces M (a;n′′), where n′′ is obtained from n by removing

three consecutive 12 -entries and inserting a 3

2 in their place.(sy|Lag) Products of moduli spaces


where the concatenation of the nj gives n with one consecutive pair of 12 -entries

removed and a 1 inserted in their place.(sy|sy) Products of moduli spaces

M sy(a; b)/R ×∏bj∈b

(M sy(bj ; cj)/R ×

∏cjk∈cj

M (cjk;njk)),

where n equals the concatenation of the njk, and all but one of the M sy(bj ; cj) aretrivial strips over the Reeb chords bj.

Proof. — This is a consequence of Theorem 10.6. The descriptions of the boundarysegments are analogous to the boundary phenomena of Proposition 6.8. At a type(Lag|Lag)1 corner we have two pairs of switches colliding. Local coordinates in themoduli space around this configuration can be taken as the lengths of the correspond-ing short boundary segments, which is a product of two half-open intervals. At a type

J.É.P. — M., 2017, tome 4


....

(sy|sy)

(sy)

(sy|Lag)(Lag)

(Lag|Lag)2

(sy)

(sy|Lag)(Lag)

(Lag|Lag)1

(Lag|Lag)2

Figure 6.10. The 2-dimensional moduli space M (a;n) and the im-mersed curve π(δM (a;n)).

(Lag|Lag)2 corner there are likewise two short boundary segments that give localcoordinates on the moduli space, see Figure 6.7. At a type (sy|Lag) corner the twoparameters are the length of the short boundary segment and the gluing parameterfor the two-level curve. Finally, at a type (sy|sy) corner the two parameters are thetwo gluing parameter for the three-level curve.

We next give alternate interpretations of the boundary phenomena in Propo-sition 6.10. Recall the notation M ∗(a;n) for the moduli space correspondingto M (a;n) in which the disks have an additional free marked point ∗ on theboundary. It comes with an evaluation map ev : M ∗(a;n) → L and a pro-jection π : M ∗(a;n)→M (a;n) forgetting the marked point, and we denoteδM (a;n) = ev−1(K).

Proposition 6.11. — If a is a Reeb chord of degree |a| = 2 and if all entries n equal 12 ,

then there is a natural orientation preserving identification between δM (a;n) andM (a;n′′), where n′′ is obtained from n by inserting in n a new entry equal to 1 atthe position given by the marked point.

The moduli space δM (a;n) ⊂M ∗(a;n) is an embedded curve with boundary. Itsboundary consists of transverse intersections with the boundary of M ∗(a;n), cor-responding to degenerations of type (sy|Lag) and (Lag|Lag)1 involving the markedpoint ∗, and to points in the interior of M ∗(a;n), corresponding to degenerations oftype (Lag|Lag)2 involving the marked point ∗.

The projection π(δM (a;n)) ⊂ M (a;n) is an immersed curve with boundary andtransverse self-intersections. Its boundary consists of transverse intersections with theboundary of M ∗(a;n). See Figure 6.10.

Proof. — This is a consequence of Theorem 10.9. Here is a sketch. The proof ofthe first statement is analogous to that of Proposition 6.9, looking at Taylor and

J.É.P. — M., 2017, tome 4


Fourier expansions. That δM (a;n) is an embedded curve with boundary follows fromtransversality of the evaluation map ev : M ∗(a;n)→ L to the knotK, which holds forgeneric almost complex structure. More refined transversality arguments show thatthe projection π(δM (a;n)) is an immersed curve with transverse self-intersectionscorresponding to holomorphic disks that meet the knot twice at non-corner points ontheir boundary.

For the other statements, note that each stratum of δM (a;n) corresponds to amoduli space M (a;n′), where n′ is obtained from n by inserting an entry 1 cor-responding to the marked point ∗. It follows from Proposition 6.10 that boundarypoints of δM (a;n) correspond to degenerations of types (sy|Lag), (Lag|Lag)1 and(Lag|Lag)2 involving the point ∗. The first two correspond to transverse intersectionsof π(δM (a;n)) with boundary strata of M (a;n) of types (sy) and (Lag), respec-tively. A dimension argument shows that degenerations of type (Lag|Lag)2 involvingthe point ∗ cannot meet the boundary of M ∗(a;n), so they correspond to boundarypoints of δM (a;n) in the interior of M ∗(a;n). They appear in pairs correspondingto holomorphic disks in which the marked point ∗ has approached a corner from theleft or right to form a new corner of weight 3/2. In δM (a;n) the two configurationson a pair are distinct (formally, they are distinguished by the position of the markedpoint ∗ on the 3-punctured constant disk attached at the weight 3/2 corner), so theygive actual boundary points. In the projection π(δM (a;n)) the two configurationbecome equal and thus give an interior point, hence π(δM (a;n)) has no boundarypoints in the interior of M (a;n).

See Section 9.6 for a discussion of orientations of the moduli spaces involved inthese arguments.

6.6. The chain map. — We can summarize the description of the moduli spaces ofpunctured holomorphic disks with switching boundary conditions in the precedingsubsections as follows. For all Reeb chords a and all integers ` > 0 the compactifiedmoduli spaces

M `(a) := M (a; 12 , . . . ,

12︸︷︷︸

2`

)

are compact oriented manifolds with boundary and corners of dimension |a| whosecodimension 1 boundaries satisfy the relations

(6.5) ∂M `(a) = M `(∂Λa) ∪ −δM `−1(a),

where ∂Λa = ∂sya and δM `−1(a) is the closure in M `−1(a) of the set

δM`−1(a) := δM (a; 12 , . . . ,

12︸︷︷︸

2`−2

)

introduced in Proposition 6.11. Again we refer to Section 9.6 for a description of theorientations involved.

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Proposition 6.12. — There exist smooth triangulations of the spaces M `(a) andgeneric chains of broken strings

Φ`(a) : M `(a) −→ Σ`

(understood as singular chains by summing up their restrictions to the simplices ofthe triangulations) satisfying the relations

(6.6) ∂Φ`(a) = Φ`(∂Λa)− (δQ + δN )Φ`−1(a).

Proof. — The idea of the proof is very simple: After connecting the end points of a tothe base point x0 by capping paths, a suitable parametrization (explained below) ofthe boundary of a holomorphic disk u ∈M`(a) determines a broken string ∂(u) ∈ Σ`.Thus we get maps

Φ`(a) : M`(a) −→ Σ`, u 7−→ ∂(u)

and the relations (6.6) should follow from (6.5). However, the map Φ`(a) in generaldoes not extend to the compactification M ` as a map to Σ` because on the boundarysome Q- or N -string can disappear in the limit. We will remedy this by suitablymodifying the maps Φ`(a) near the boundaries (inserting spikes).

Before doing this, let us discuss parametrizations of the broken string ∂(u) foru ∈M`(a). Near a switch we can pick holomorphic coordinates on the domain (withvalues in the upper half-disk) and the target (provided by Lemma 8.6) in whichthe normal projection of u consists of two holomorphic functions near a corner as inSection 4. The discussion in that section shows that in these coordinates ∂(u) satisfiesthe matching conditions on the m-jets required in the definition a broken string.We take near each corner a parametrization of ∂(u) induced by such holomorphiccoordinates and extend them arbitrarily away from the corners to make ∂(u) a brokenstring in the sense of Definition 5.1. Note that the space of such parametrizations iscontractible.

Now we proceed by induction over |a| = 0, 1, 2.

Case |a| = 0. — In this case M `(a) consists of finitely many oriented points and weset Φ`(a)(u) := ∂(u) (picking a parametrization of the boundary as above).

Case |a| = 1. — We proceed by induction on ` = 0, 1, . . . . For ` = 0, on the boundary∂M 0(a) = M 0(∂Λa) we are already given the map Φ0(∂Λa). We extend it to a mapΦ0(a) : M 0(a)→ Σ0 by sending u to ∂(u) with parametrizations matching the givenones on ∂M 0(a), so that ∂Φ0(a) = Φ0(∂Λa) holds.

Now suppose that we have already defined Φ0(a), . . . ,Φ`−1(a) such that the rela-tions (6.6) hold up to ` − 1. According to (6.5), the boundary ∂M `(a) is identifiedwith the union of domains of the maps on the right hand side of (6.6). On the otherhand, on the interior M`(a) we are given the map Φ`(a) described above. Further-more, by Proposition 6.8 and Remark 8.13, elements u close to the boundary pointsin δM `−1(a) ⊂ ∂M `(a) have spikes (shrinking as u tends to the boundary) roughlyin the same direction as those on the boundary. So near ∂M `(a) we can interpolatebetween the map on the boundary given by the right hand side of (6.6) and the

J.É.P. — M., 2017, tome 4


map Φ`(a) on the interior to obtain a map Φ`(a) : M `(a)→ Σ` satisfying (6.6). Sincethe modification of Φ`(a) can be done away from the finite set δM `(a) ⊂M`(a), Φ`(a)

is a generic 1-chain of broken strings. This concludes the inductive step. Since we aredealing with 1-chains, a smooth triangulation just amounts to a parametrization ofthe components of M `(a) by intervals whose boundary points avoid the set δM `(a).

Case |a| = 2. — We proceed again by induction on ` = 0, 1, . . . . For ` = 0, we againdefine Φ0(a) : M 0(a) → Σ0 by sending u to ∂(u), with parametrizations matchingthe given ones on ∂M 0(a), so that ∂Φ0(a) = Φ0(∂Λa) holds.

Now suppose that we have already defined Φ0(a), . . .Φ`−1(a) and triangulationsof their domains such that they are generic 2-chains of broken strings and the re-lations (6.6) hold up to ` − 1. As in the case of 1-chains, the boundary ∂M `(a) isidentified via (6.5) with the union of domains of the right hand side of (6.6), so asbefore we define Φ`(a) on that boundary via the maps Φ`(∂Λa) resp. δΦ`−1(a). By in-duction hypothesis, these maps coincide at corner points. Note that the map δΦ`−1(a)

inserts spikes at the intersection points with the knot. According to Proposition 6.10and Remark 8.13, elements u close to the codimension one boundary strata δM `−1(a)

have spikes roughly in the same direction as those on the boundary (shrinking in sizeas u tends to the boundary). Elements close to a corner point where two boundarystrata of δM `−1(a) meet have two spikes roughly in the same directions as those onthe nearby boundary strata (which both shrink as u tends to the corner point), seeRemark 8.13. So we can interpolate between the given map on the boundary ∂M `(a)

and the map Φ`(a) on the interior M`(a) to obtain a map Φ`(a) : M `(a) → Σ`

satisfying (6.6).Recall that δM `(a) is an immersed 1-dimensional submanifold with finitely many

transverse self-intersections in the interior, and which meets the boundary transverselyaway from the corners. The modification of Φ`(a) can be done away from the finiteset of self-intersections of δM `(a) in the interior. Moreover, the modification of Φ`(a)

near the boundary only involves inserting spikes at switching points of broken strings,which can be performed away from the finitely many interior intersection points ofthe broken strings with the knot and thus does not affect δM `(a).

We pick a smooth triangulation of M `(a) transverse to δM `(a) (i.e., transverse toits 1-dimensional strata as well as its self-intersection points) and inducing the giventriangulation on the boundary. By the discussion in the preceding paragraph, Φ`(a)

(interpreted as the sum over its restriction to simplices) is a generic 2-chain of brokenstrings. This concludes the inductive step and thus the proof of Proposition 6.12.

Given a Reeb chord a, we define

Φ(a) :=

κ∑`=0

Φ`(a) ∈ C(Σ) =∞⊕=0

C(Σ`).

Here κ is the constant from the Finiteness Theorem 6.5. The relation (6.6) for thechains Φ`(a) translates into

(6.7) ∂Φ(a) = Φ(∂sya)− δΦ(a), δ = δQ + δN .

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Given a d-simplex of Reeb strings a = α1a1 . . . amαm+1 : ∆→ Rm we define

Φ(a) := α1Φ(a1) . . . αmΦ(am)αm+1 ∈ C(Σ).

Here the boundary arcs are concatenated in the obvious way to obtain broken strings.For singular simplices ∆i appearing as domains in Φ(ai), the corresponding term inΦ(a) has by our orientation convention the domain

∆×∆1 × · · · ×∆m

in this order of factors.

Theorem 6.13. — The map Φ is a chain map from (C∗(R), ∂Λ) to (C∗(Σ), ∂+δQ+δN ).

Proof. — Using (6.7) we compute for a ∈ Cd(R) as above, with ∗ = d+ |a1|+ · · ·+|ai−1|:

∂Φ(a) = Φ(∂singa) +

m∑i=1

(−1)∗α1Φ(a1)α2 · · · ∂Φ(ai) · · ·αmΦ(am)αm+1

= Φ(∂singa) +

m∑i=1

(−1)∗α1Φ(a1)α2 · · ·(

Φ(∂syai)− δΦ(ai))· · ·αm+1

= Φ(∂singa) + Φ(∂sya)− δΦ(a).

Since ∂Λ = ∂sing + ∂sy, this proves the theorem.

Compatibility with length filtrations. — Holomorphic disks with switching boundaryconditions have a length decreasing property that leads to the chain map Φ respect-ing the length (or action) filtration, which is central for our isomorphism proof. Letu ∈ M (a;n) be a holomorphic disk with k boundary segments that map to Q. Letσ1, . . . , σk be the corresponding curves in Q and let L(σi) denote the length of σi.Recall that the Reeb chord a is the lift of a binormal chord on the link K and thatthe action

∫apdq of a equals the length of the underlying chord in Q, which we write

as L(a). In Section 8.2 we utilize the positivity of a scaled version of the contact formon holomorphic disks to show the following result (Proposition 8.9).

Proposition 6.14. — If u ∈M (a;n) is as above thenk∑i=1

L(σi) 6 L(a),

with equality if and only if u is a trivial half strip over a binormal chord.

Recall that both chain complexes (C∗(R), ∂Λ) and (C∗(Σ), ∂ + δQ + δN ) carrylength filtrations that were defined in Sections 6.2 and 5.5, respectively. Recall alsothat the length filtration on C∗(Σ) does not count the lengths of Q-spikes. Hence theinsertion of Q-spikes in the definition of the chain map Φ does not increase lengthand Proposition 6.14 implies

Corollary 6.15. — The chain map Φ in Theorem 6.13 respects the length filtrations,i.e., it does not increase length.

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7. Proof of the isomorphism in degree zero

In the previous section we have constructed a chain map Φ : (C∗(R), ∂Λ) →(C∗(Σ), D), where D = ∂ + δQ + δN . In this section we finish the proof of Theo-rem 1.2 by showing that the induced map Φ∗ : H0(R, ∂Λ) → H0(Σ, D) in degreezero is an isomorphism. Whereas the results in the previous section hold for any3-manifold Q with a metric of nonpositive curvature which is convex at infinity, inthis section we need to restrict to the case Q = R3 with its Euclidean metric. Thisrestriction will allow us to obtain crucial control over the straightening procedure forQ-strings described in Proposition 7.6 (see the comment in Remark 7.9 below).

As a first step, we will slightly extend the definition of broken strings to includepiecewise linear Q-strings. A relatively simple approximation result will show thatthe inclusion of broken strings with piecewise linear Q-strings into all broken stringsinduces an isomorphism on string homology in degree 0.

The central piece of the argument will then consist of deforming the complex ofbroken strings with piecewise linear Q-strings into the subcomplex of those with linearQ-strings.

It is important that both of these reduction steps can be done preserving the lengthfiltration on Q-strings. The final step of the argument then consists of comparing thecontact homologyH0(R, ∂Λ) with the homology of the chain complex of broken stringswith linear Q-strings. At this stage, we will use the length filtrations to reduce to thecomparison of homology in degrees 0 and 1 in small length windows containing atmost one critical value.

7.1. Approximation by piecewise linear Q-strings. — In the following we enlargethe space of broken Cm-strings Σ, keeping the same notation, to allow for Q-strings tobe piecewise Cm. Here a Q-string s2i : [a2i−1, a2i]→ Q is called piecewise Cm if thereexists a subdivision a2i−1 = b0 < b1 < · · · < br = a2i such that the restriction of s2i

to each subinterval [bj−1, bj ] is Cm. For a generic d-chain S : ∆d → Σ` (d = 0, 1, 2) werequire that the number of subdivision points on each Q-string is constant over thesimplex ∆d. The subdivision points can vary smoothly over Σd but have to remaindistinct. If for some subdivision point bj the two Cm-strings meeting at bj fit togetherin a Cm-fashion for all λ ∈ ∆d, then we identify S with the generic d-chain obtainedby removing the subdivision point bj .

We allow Q-strings in a generic d-chain S to meet the knot K at a subdivisionpoint bj , provided at such a point the derivatives from both sides satisfy the gener-icity conditions in Definition 5.3. If this occurs for some parameter value λ∗ ∈ ∆2

in a generic 2-chain, then we require in addition that the corresponding Q-stringmeets K at the subdivision point bj(λ) for all λ in the component of λ∗ in the do-main MδQ of δQS defined in Section 5.3. These conditions ensure that the operatorD = ∂ + δQ + δN extends to generic chains of piecewise Cm strings satisfying therelations in Proposition 5.8.

J.É.P. — M., 2017, tome 4


The subspace Σpl ⊂ Σ of broken strings whose Q-strings are piecewise linear giverise to an inclusion of a D-subcomplex

(7.1) C∗(Σpl) ipl−−−→ C∗(Σ).

For this to hold, we choose the Q-spikes inserted under the map δN to be degenerate3-gons, i.e., short segments orthogonal to the knot traversed back and forth. ThenC∗(Σpl) becomes a D-subcomplex.

We will also consider the subspace Σlin ⊂ Σpl of broken closed strings whose Q-strings are (essentially) linear : any two points x1, x2 ∈ K determine a unique linesegment [x1, x2] in R3 connecting them. For technical reasons, special care has tobe taken when such a linear Q-string becomes very short. Indeed, near the diagonal∆ ⊂ K × K we deform the segments to piecewise linear strings with one corner insuch a way that at each point of the diagonal, instead of a segment of length zero wehave a degenerate 3-gon as above, i.e., a short spike in direction of the curvature ofthe knot (which we assume vanishes nowhere). Now Σlin ⊂ Σpl consists of all brokenclosed strings whose Q-strings are constant speed parametrizations of such (possiblydeformed) segments. In this way,

(7.2) C∗(Σlin)ilin−→ C∗(Σpl)

will be an inclusion of a D-subcomplex.Recall from Section 5.5 that these complexes are filtered by the length L(β), i.e.,

the maximum of the total length of Q-strings over all parameter values of the chain,where in the length we do not count Q-spikes. With these notations, we have thefollowing approximation result.

Proposition 7.1. — There exist maps

F0 : C0(Σ) −→ C0(Σpl), F1 : C1(Σ) −→ C1(Σpl)

andH0 : C0(Σ) −→ C1(Σ), H1 : C1(Σ) −→ C2(Σ)

satisfying with the map ipl from (7.1):(i) F0ipl = 1l and DH0 = iplF0 − 1l;(ii) F1ipl = 1l and H0D +DH1 = iplF1 − 1l;(iii) F0, H0, F1 and H1 are (not necessarily strictly) length-decreasing.

Proof. — We first define F0 and H0. Given β ∈ C0(Σ), we pick finitely many subdi-vision points pi on the Q-strings in β (which include all end points) and define H0β

to be the straight line homotopy from β to the broken string F0β whose Q-strings arethe piecewise linear strings connecting the pi. We choose the subdivision so fine thatthe Q-strings in H0β remain transverse to K at the end points and do not meet K inthe interior. The N -strings are just slightly rotated near the end points to match the

J.É.P. — M., 2017, tome 4


new Q-strings, without creating intersections with K. Then H0β is a generic 1-chainin Σ satisfying

∂H0β = F0β − β, δQH0β = δNH0β = 0.

If β is already piecewise linear we include the corner points in the subdivision toensure F0β = β, so that condition (i) holds.

To define F1 and H1, consider a generic 1-simplex β : [0, 1] → Σ. We pick finitelymany smooth paths of subdivision points pi(λ) on the Q-strings in β(λ) (which in-clude all end points) and define H1β to be the straight line homotopy from β to the1-simplex F1β whose Q-strings are the piecewise linear strings connecting the pi(λ).Here we choose the pi(λ) to agree with the ones in the definition of H0 at λ = 0, 1 aswell as at the finitely many values λj where some Q-string intersects the knot in itsinterior (so at such λj the intersection point with K is included among the pi(λj)).Note that for this we may first have to add new subdivision points on the Q-stringson β(λ) for λ = 0, 1, λj , which is allowed due to the identification above. Moreover,we choose the subdivision so fine that the Q-strings in H1β remain transverse to Kat the end points and meet K in the interior exactly at the values λj above. TheN -strings are just slightly rotated near the end points to match the new Q-strings,without creating new intersections with K besides the ones already present in β thatare continued along the homotopy. Then H1β is a generic 2-chain in Σ satisfying

(∂H1 + H0∂)β = F1β − β, (δQH1 + H0δQ)β = (δNH1 + H0δN )β = 0.

If β is already piecewise linear we include the corner points in the subdivision toensure F1β = β, so that condition (ii) holds.

7.2. Properties of triangles for generic knots. — In our arguments, we will as-sume that the knot K is generic. In particular, we will use that it has the propertieslisted in the following lemma.

Lemma 7.2. — A generic knot K ⊂ R3 has the following properties:(i) There exists an S ∈ N such that each plane intersects K at most S times.(ii) The set T ⊂ K of points whose tangent lines meet the knot again is finite (and

each such tangent line meets the knot in exactly one other point).

Proof. — We prove part (i). For a generic knot K parametrized by

γ : S1 = R/LZ −→ R3,

the first four derivatives (γ, γ(2), γ(3), γ(4)) span R3 at each t ∈ S1. (For this, use thejet transversality theorem [25, Chap. 3] to make the corresponding map S1 → (R3)4

transverse to the codimension two subset consisting of quadruples of vectors that liein a plane.) It follows that there exists an ε > 0 such that γ meets each plane at mostfour times on a time interval of length ε. (Otherwise, taking a limit of quintuples oftimes mapped into the same plane whose mutual distances shrink to zero, we wouldfind in the limit an order four tangency of γ to a plane, which we have excluded.)Hence γ can meet each plane at most 4L/ε times.

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x1 = γ(s)

v2

x2

v3

x3 = γ(r)

(1− u)x2 + ux3

(1− t)x1 + t((1− u)x2 + ux3)

Figure 7.1. Parametrization of a triangle.

The proof of part (ii) is contained in the proof of Lemma 7.10(b) below. It relieson choosing K such that its curvature vanishes nowhere.

Now we consider the space of triangles in R3 with pairwise distinct cornersx1, x2, x3 such that x1 and x3 lie on the knot K. Using an arclength parametrizationγ : S1 = R/LZ→ K we identify this space with the open subset

T = (s, x2, r) ∈ S1 × R3 × S1 | x1 = γ(s), x2, x3 = γ(r) are distinct.

We parametrize each triangle [x1, x2, x3] by the map (see Figure 7.1)

[0, 1]2 −→ R3, (u, t) 7−→ (1− t)x1 + t((1− u)x2 + ux3

).

Lemma 7.3. — For a generic 1-parameter family of triangles β : [0, 1] → T , λ 7→(sλ, xλ2 , r

λ) the following holds.(a) The evaluation map

evβ : [0, 1]3 −→ R3, (λ, u, t) 7−→ (1− t)xλ1 + t((1− u)xλ2 + uxλ3

)is transverse to K on its interior, where we have set xλ1 = γ(sλ) and xλ3 = γ(rλ).

(b) The map (λ, u) 7→ ∂evβ∂t (λ, u, 0) meets the tangent bundle to K transversely in

finitely many points. At these points the triangle is tangent to the knot at xλ1 but notcontained in its osculating plane.

(c) The points in (b) compactify the set ev−1β (K) ∩ [0, 1]2 × (0, 1] to an embedded

curve in [0, 1]3 transverse to the boundary. Its image in [0, 1]2 under the projection(λ, u, t) 7→ (λ, u) is an immersed curve with transverse self-intersections.

Proof. — Part (a) follows from standard transversality arguments. For part (b) weintroduce

v2 := x2 − x1, v3 := x3 − x1, ν :=v2 × v3

|v2 × v3|.

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Thus v2, v3 are tangent to the sides of the triangle at x1 and ν is a unit normal vectorto the triangle. So the space of triangles that are tangent to the knot at x1 is the zeroset of the map

F : T −→ R, (s, x2, r) 7−→ 〈γ(s), ν〉 =〈v2, v3 × γ(s)〉|v2 × v3|

.

The last expression shows that along the zero set the variation of F in direction x2 (orequivalently v2) is nonzero provided that v3 × γ(s) 6= 0. So F−1(0) is a transverselycut out hypersurface in T outside the set T0 where v3 = γ(r) − γ(s) and γ(s) arecollinear. By Lemma 7.2(ii) the set T0 has codimension 2. Hence a generic curveβ : [0, 1] → T avoids the set T0 and intersects F−1(0) transversely, which impliesthe first statement in (b). The second statement in (b) follows similarly from the factthat the set of triangles contained in the osculating plane at x1 has codimension 2

in T .For part (c), consider a point (λ0, u0) as in (b). To simplify notation, let us shift

the parameter interval such that λ0 = 0 is an interior point. Then with the obviousnotation νλ etc the following conditions hold at λ = 0:

a := 〈γ(s0), ν0〉 = 0, b := 〈γ(s0), ν0〉 6= 0, c :=d

dλ

∣∣∣λ=0〈γ(sλ), νλ〉 6= 0.

Here the first condition expresses the fact that the triangle is tangent to the knot at x01,

the second on that the triangle is not contained in the osculating plane, and the thirdone the transversality of the map in (b) to the tangent bundle of K. Intersectionsof K with triangles β(λ) for λ close to zero can be written in the form γ(sλ + s) withs = O(λ) and must satisfy the equation

0 =⟨γ(sλ + s)− γ(sλ), νλ

⟩=⟨sγ(sλ) +

1

2s2γ(sλ) +O(s3), νλ

⟩.

Ignoring the trivial solution s = 0, we divide by s and obtain using s = O(λ):

0 =⟨γ(sλ) +

1

2sγ(sλ) +O(s2), νλ

⟩=⟨γ(s0) + λγ(s0) +

1

2sγ(s0) +O(λ2), ν0 + λν0 +O(λ2)

⟩= 〈γ(s0), ν0〉+ λ

[〈γ(s0), ν0〉+ 〈γ(s0), ν0〉+O(λ)

]+ s[1

2〈γ(s0), ν0〉+O(λ)

]= a+ λ

[b+O(λ)

]+ s[1

2c+O(λ)

].

Since a = 0 and and b, c are nonzero, this equation has for each λ a unique solution sof the form

s = −2b

cλ+O(λ2).

Now recall that by hypothesis γ(s0) is a multiple of (1−u0)v02 +u0v0

3 . If it is a positive(resp. negative) multiple, then only solutions with s > 0 (resp. s < 0) will lie in thetriangle. So in either case the solutions describe a curve with boundary and part (c)follows.

J.É.P. — M., 2017, tome 4


Remark 7.4. — Lemma 7.3 shows that, given a generic 1-parameter family of trian-gles β : [0, 1] → T , the associated 2-parameter family (λ, u) 7→ evβ(λ, u, ·) can bereparametrized in t to look like the Q-strings in a generic 2-chain of broken strings.To see the last condition (2e) in Definition 5.3, consider a parameter value (λ, u) asin Lemma 7.3(b). Since the triangle is not contained in the osculating plane at xλ1 ,the linear string t 7→ evβ(λ, u, t) deviates quadratically from the knot, so its projec-tion normal to the knot has nonvanishing second derivative at t = 0. Hence we canreparametrize it to make its second derivative vanish and its third derivative nonzeroas required in condition (2e). We will ignore these reparametrizations in the following.

Remark 7.5. — Lemma 7.3 remains true (with a simpler proof) if in the definitionof the space of triangles T we allow x3 to move freely in R3 rather than only on theknot; this situation will also occur in the shortening process in the next subsection.

Let us emphasize that in the space T we require the points x1, x2, x3 to be dis-tinct. Now in a generic 1-parameter family of triples (x1, x2, x3) with x1, x3 ∈ K thepoints x1, x3 may meet for some parameter values, so this situation is not covered byLemma 7.3. See Remark 7.7 below on how to deal with this situation.

7.3. Reducing piecewise linear Q-strings to linear ones. — In this subsection wedeform chains in Σpl to chains in Σlin, not increasing the length of Q-strings in theprocess. The main result of this subsection is

Proposition 7.6. — For a generic knot K there exist maps

F0 : C0(Σpl) −→ C0(Σlin), F1 : C1(Σpl) −→ C1(Σlin)

andH0 : C0(Σpl) −→ C1(Σpl), H1 : C1(Σpl) −→ C2(Σpl)

satisfying with the map ilin from (7.2):(i) F0ilin = 1l and DH0 = ilinF0 − 1l;(ii) F1ilin = 1l and H0D +DH1 = ilinF1 − 1l;(iii) F0, H0, F1 and H1 are (not necessarily strictly) length-decreasing.

Proof. — We assume that K satisfies the genericity properties in Section 7.2. We firstconstruct the maps H0 and F0.

For each simplex β ∈ Cpl0 (Σ) we denote byM(β) the total number of corners in the

Q-strings of β, not counting the corners in Q-spikes (which are by definition 3-gons).Connecting each corner to the starting point of its Q-string, we obtainM(β) trianglesconnecting the various Q-strings to the segments between their end points. We definethe complexity of β ∈ Cpl

0 (Σ) to be the pair of nonnegative integers

c(β) := (M(β), I(β)),

where I(β) is the number of interior intersection points of the first triangle with K(we set I(β) = 0 in the case M(β) = 0, i.e., if there are no triangles). Note thatby part (i) of Lemma 7.2 we know that I is bounded a priori by a fixed constant

J.É.P. — M., 2017, tome 4


S = S(K). We define the maps H0 and F0 by induction on the lexicographical orderon complexities c(β). For c(β) = (0, 0) we set F0β = β and H0β = 0.

For the induction step, let β ∈ Cpl0 (Σ) be a 0-simplex and assume that F0 and H0

satisfying (i) and (iii) have been defined for all simplices of complexities c < c(β).Let the first triangle of β have vertices x1, x2, x3, where x1 is the starting point ofthe first Q-string which is not a segment, and x2 and x3 are the next two cornerson that Q-string (x3 might also be the end point). Since there are only finitely manyintersections of the knot K with the interior of the triangle (and none with its sides),we can find a segment connecting x2 to a point x′3 on the segment x1x3 which is soclose to x3 that the triangle x2x

′3x3 does not contain any intersection points with the

knot. Let hβ ∈ Cpl1 (Σ) be the 1-simplex obtained by sweeping the first triangle by

the family of segments from x1 to a varying point (1 − u)x2 + ux′3 on the segment[x2, x

′3], followed by the segment from that point to x3 and the remaining segments

to x4 etc. See Figure 7.2 (the point y and the shaded region play no role here andare included for later use). The N -string ending at x1 (and if there is one, also the

x2

β

x1

x′3

x3

x4

(1− u)x2 + ux′3

fβ

y

Figure 7.2. Reducing the number of corner points.

N -string starting at x3) is “dragged along” without creating intersections with K,and all remaining N -and Q-strings remain unchanged in the process.

The 1-simplex hβ has boundary ∂(hβ) = β′ − β, where β′ is the 0-simplex at theend of the sweep with first segment [x1, x3]. We define

fβ := Dhβ + β = β′ + δQhβ + δNhβ.

By construction we have δNhβ = 0 and M(β′) < M(β), hence c(β′) < c(β). Thedomain of δQhβ consists of those finitely many points where the triangle intersects Kin its interior, so that δQhβ consists of broken strings with one more Q-string (which islinear) and with the same total number of corners as β. But since the new first triangleis contained in the original first triangle for β, and one of the intersection points isnow the starting point of the new Q-string, we have I(δQhβ) < I(β). Altogether wesee that c(fβ) < c(β), so by induction hypothesis F0 and H0 are already definedon fβ. We set

F0β := F0fβ and H0β := H0fβ + hβ

J.É.P. — M., 2017, tome 4


and verify that indeed (using condition (i) on fβ)

DH0β = DH0fβ +Dhβ = F0fβ − fβ + fβ − β = F0β − β,

so condition (i) continues to hold. Condition (iii) holds by induction hypothesis in viewof L(fβ) 6 L(β) and L(hβ) 6 L(β). Since every β ∈ Cpl

0 (Σ) has finite complexity,this finishes the definition of F0 and H0.

We next construct the maps H1 and F1, following the same strategy. For this, wefirst extend the notion of complexity c = (M, I) to 1-chains with piecewise linearQ-strings. For a 1-simplex β : [0, 1]→ Σpl, we set

M(β) := maxλ∈[0,1]

M(β(λ)), I(β) := maxλ∈[0,1]

I(β(λ)).

Note that I(β) is still bounded by the constant S = S(K) in Lemma 7.2. Notealso that, according to our definition of chains of piecewise linear strings, the numberM(β(λ)) of corner points ofQ-strings in β(λ) is actually constant equal to the maximalnumber M(β). Observe that with this definition of complexity for 1-chains, the mapsh0 := h and H0 used in the argument for 0-chains do not increase complexity.

Again our definition of F1 and H1 proceeds by induction on the lexicographic orderon complexity. For simplices β ∈ Cpl

1 (Σ) with M = 0 we set F1β = β + H0Dβ andH1β = 0. Then (ii) holds by construction, and (iii) holds since H0 and D are length-decreasing.

For the induction step, let β ∈ Cpl1 (Σ) be a 1-simplex, and assume that F1 and H1

satisfying (ii) and (iii) have been defined for all 1-simplices of complexity c < c(β).Using a parametrized version of sweeping the first triangle, we obtain a 2-chain h1β ∈Cpl

2 (Σ). By construction its boundary satisfies ∂h1β+ h0∂β = β′− β, where β′ is the1-simplex at the end of the sweep with first segment [x1, x3], see Figure 7.3. We now

0

h1∂β

1

u

β

h1∂β

β′

Zβ

δNβ

δNh1β

δNh1β

δQβ δQβ

δQh1β

δQh1βδQh1∂β

δQh1∂βδQh1β

δQβ′

Figure 7.3. The domain of h1β.

J.É.P. — M., 2017, tome 4


definef1β : = Dh1β + h0Dβ + β

= β′ + (δQh1 + h0δQ)β + (δNh1 + h0δN )β.

We claim that c(f1β) < c(β). To see this, we need to show that the three terms onthe right hand side of the last displayed equation have complexity lower that c(β).For β′ this holds because its Q-strings have one fewer corner, i.e., M(β′) < M(β).The domain of (δQh1 + h0δQ)β consists of the finitely many curves in which the firsttriangle intersects K at an interior point y, so that (δQh1 +h0δQ)β consists of brokenstrings with one more Q-string (which is linear) and with the same total numberof corners as β. But since the new first triangle (the shaded region in Figure 7.2)is contained in the original first triangle for each parameter value in β, and oneof the intersection points is now the starting point of the new Q-string, we haveI((δQh1 + h0δQ)β) < I(β). The domain of (δNh1 + h0δN )β consists of the finitelymany straight line segments [u, 1]× λ emanating from the parameter values (u, λ)

corresponding to the tangencies of the triangle [x1, x2, x3] to the knot at x1, seeFigure 7.3 where one such point of tangency is shown as Zβ. So (δNh1 + h0δN )β

consists of broken strings with one more Q-spike and with the same total numberof corners as β. But since the new triangle with corners x1, (1 − u)x2 + ux′3, x3 iscontained in the original first triangle at parameter value λ, and one of the intersectionpoints with the knot is the corner point x1 of the new triangle (which does not counttowards I), we have I((δQh1 + h0δQ)β) < I(β) and the claim is proved.

According to the claim, F1 and H1 are defined on f1β and we set

F1β := F1f1β and H1β := H1f1β + h1β.

To distinguish the proposed extensions from the maps given by induction hypothesis,we temporarily call the extended versions H1 and F1, so we can write

F1 := F1f1 and H1 := H1f1 + h1

without ambiguity. Recall also that in this notation H0 = H0f0 + h0. Now usingf1 = h0D +Dh1 + 1l we compute

DH1 + H0D = DH1f1 +Dh1 + H0f0D + h0D

= (F1f1 − f1 −H0Df1) + (f1 − 1l− h0D) + H0f0D + h0D

= F1 − 1l + H0(f0D −Df1).

Using f1 = h0D + Dh1 + 1l again and f0 = Dh0 + 1l, we find Df1 = Dh0D + D =

(Dh0 + 1l)D = f0D, so that the last term in the displayed equation vanishes and theextensions H1,F1 have the required properties. This completes the induction stepand hence the proof of Proposition 7.6.

Remark 7.7. — If in a 1-simplex β as in the preceding proof the third point x3 of thefirst triangle is the end point of the corresponding Q-string and thus constrained tolie on the knot, then the points x1 and x3 can cross each other for some parametervalues λ in the chain. The homotopy h1β then shrinks the corresponding degenerate

J.É.P. — M., 2017, tome 4


triangle at parameter λ to a constant Q-string, which according to our conventionfrom Section 7.1 we interpret as a linear Q-spike in the direction of the degeneratetriangle. Incidentally, the segment [x2, x3] is always short throughout the shorteningprocess, so if x1 and x3 agree then the triangle is already a linear Q-spike withoutfurther shrinking.

Remark 7.8. — Definition 5.4 implies that if a Q-string in β in the preceding proofis a (piecewise linear) Q-spike, then it never intersects the knot in its interior andremains a Q-spike throughout the shortening process (which ends with a degeneratetriangle as in Remark 7.7). This property ensures that H0 and H1 indeed do notincrease length, which does not count Q-spikes.

Remark 7.9. — The proof relies crucially on the (trivial) fact that the new triangle[y, (1 − u)x2 + ux′3, x3] (the shaded region in Figure 7.2) obtained by splitting theQ-string at an intersection point y with K is contained in the old triangle [x1, x2, x3].This is the only place where we use that the metric is Euclidean; the rest of the proofworks equally well for any metric of nonpositive curvature.

7.4. Properties of linear Q-strings for generic knots. — Now we consider thespace of 2-gons, i.e., straight line segments starting and ending on the knot. Thisspace is canonically identified with K ×K by associating to each 2-gon its endpointson K. We consider the squared distance function

E : K ×K −→ R, E(x, y) =1

2|x− y|2.

Lemma 7.10. — For a generic knot K ⊂ R3 the following holds for the space K ×Kof 2-gons (see Figure 7.4).

0 L

L

s

t

K ×K

−∇ESQ

SQ

Figure 7.4. The space of 2-gons.

(a) E attains its minimum 0 along the diagonal, which is a Bott nondegeneratecritical manifold; the other critical points are nondegenerate binormal chords of index0, 1, 2.

J.É.P. — M., 2017, tome 4


KK

p

pξ pη

γ(s)q

P

Q

`ξ,η

Figure 7.5. A 2-gon becoming tangent to K at an endpoint.

(b) The subset SQ ⊂ K×K of 2-gons meeting K in their interior is a 1-dimensionalsubmanifold with boundary consisting of finitely many 2-gons tangent to K at oneendpoint, and with finitely many transverse self-intersections consisting of finitelymany 2-gons meeting K twice in their interior.

(c) The negative gradient −∇E is not pointing into SQ at the boundary points.

Proof

(a) In terms of an arclength parametrization γ of K we write the energy as afunction E(s, t) = 1

2 |γ(s)− γ(t)|2. We compute its partial derivatives

(7.3)

∂E

∂s= 〈γ(s)− γ(t), γ(s)〉, ∂E

∂t= 〈γ(t)− γ(s), γ(t)〉,

∂2E

∂s2= |γ(s)|2 + 〈γ(s)− γ(t), γ(s)〉, ∂2E

∂s∂t= −〈γ(s), γ(t)〉,

∂2E

∂t2= |γ(t)|2 + 〈γ(t)− γ(s), γ(t)〉.

We see that critical points of E are points on the diagonal s = t and binormal chords(where s 6= t), and the Hessian of E at s = t equals

(1 −1−1 1

). Its kernel is the tangent

space to the diagonal and it is positive definite in the transverse direction. This provesBott nondegeneracy of the diagonal. Nondegeneracy of the binormal chords is achievedby a generic perturbation of K.

(b) We choose K so that its curvature is nowhere 0 (which holds generically). Thenthere exists δ > 0 such that no 2-gon of positive length < δ intersects the knot inan interior point. Consider the tangential variety τK of K (where γ : [0, L]→ R3 is aparametrization of K)

τK := γ(s) + rγ(s) | s ∈ [0, L], r ∈ R ⊂ R3.

Since the curvature of K is nowhere zero, there exists δ > 0 such that for each s theline segment γ(s) + rγ(s) | r ∈ (−δ, δ) intersects K only at r = 0. Let N(δ) denotethe union of these line segments. After small perturbation, the surface τK r N(δ)

J.É.P. — M., 2017, tome 4


intersects K transversely. This shows that there are finitely many 2-gons that aretangent to K at one endpoint and that this is a transversely cut out 0-manifold.Moreover, transversality implies that for each 2-gon that is tangent to K at oneendpoint p, the tangent line Q to K at the other endpoint q does not lie in theosculating plane P (the plane spanned by the first two derivatives of γ) at p; seeFigure 7.5.

We claim that the 2-gon [p, q] is the boundary point of a unique local embeddedcurve of 2-gons intersectingK in their interior. To see this, we choose affine coordinates(x, y, z) on R3 in which p = (0, 0, 0), q = (1, 0, 0), P is the (x, y)-plane, andQ is parallelto the z-axis. Then K can be written near p as a graph over the x-axis in the form

y = κx2 +O(x3), z = O(x3),

and near q as a graph over the z-axis in the form

x = 1 +O(z2), y = O(z2).

Here 2κ 6= 0 is the curvature of K at p, and after a further reflection we may assumethat κ > 0. We fix a small ε > 0 (to be chosen later) and consider points ξ, η on thex-axis with −ε < ξ < η < 2ε. Let pξ, pη be the points of K near p with x-coordinatesξ, η and let `ξ,η be the line through pξ and pη. Let π(x, y, z) = (x, z) be the projectiononto the (x, z)-plane. Since the line `ξ,η is close to the x-axis and K is tangent tothe z-axis at q, the projected curves π(`ξ,η) and π(K) intersect in a unique point rξ,ηin the (x, z)-plane near π(q) = (1, 0). Let fξ(η) denote the difference in the y-valuesbetween the points of K and `ξ,η lying over rξ,η. Thus fξ(η) is the “distance in they-direction” between `ξ,η and K near q. To compute the function fξ(η), note that theslope of the line through the points (ξ, κξ2) and (η, κη2) on the parabola y = κx2

equals κ(ξ+η), so the y-value of this line at x = 1 is of the form κ(ξ+η)+O(ξ2 +η2).The linear term persists for the function fξ(η), hence

fξ(η) = κ(ξ + η) +O(ξ2 + η2).

For ε sufficiently small, we see that if ξ > 0, then fξ(η) > 0 for all η ∈ (ξ, 2ε). Supposetherefore that ξ < 0. Then for ε sufficiently small we have fξ(0) = κξ + O(ξ2) < 0,fξ(−2ξ) = −κξ + O(ξ2) > 0, and f ′ξ(η) = κ + O(|ξ| + |η|) > 0. Thus for everyξ ∈ (−ε, 0) there exists a unique η(ξ) ∈ (ξ, 2ε) such that fξ(η(ξ)) = 0, i.e., theline `ξ,η(ξ) intersects K near q. Moreover, the point η(ξ) depends smoothly on ξ

and satisfies 0 < η(ξ) < −2ξ. This shows that the 2-gons with endpoints near p, qintersecting K in their interior form a smooth curve parametrized by ξ ∈ (−ε, 0),consisting of the corresponding segments of the lines `ξ,η(ξ). As this curve extendssmoothly to ξ = 0 by the 2-gon [p, q], the claim is proved.

So we have shown that the subset SQ ⊂ K × K avoids a neighborhood of thediagonal and is a 1-manifold with boundary near the finitely many 2-gons that aretangent toK at an endpoint. Away from these sets, a generic perturbation ofK makesthe evaluation map at the interior of the 2-gons transverse to K. Since the conditionthat a chord meets K in the interior is codimension one, and the condition that the

J.É.P. — M., 2017, tome 4


tangent line at the intersection is parallel to the chord is of codimension three andcan thus be avoided for generic K, we conclude that (b) holds.

(c) Consider a boundary point of SQ, i.e., a 2-gon [p, q] tangent to K at one end-point, say at p. Let p = γ(s) and q = γ(t) for an arclength parametrization of K suchthat γ(s) is a positive multiple of q − p; see Figure 7.5. By equation (7.3) we have∂E/∂s = 〈p − q, γ(s)〉 < 0, so the parameter s strictly increases in the direction of−∇E. On the other hand, the description in (b) shows that s strictly decreases as wemove into SQ. Hence −∇E is not pointing into SQ at [p, q].

More generally, for an integer ` > 1 we consider the space (K ×K)` of `-tuples of2-gons with the energy and length functions E`, L` : (K ×K)` → R,

E`(x1, y1, . . . , x`, y`) :=1

2

∑i=1

|xi − yi|2,

L`(x1, y1, . . . , x`, y`) :=∑i=1

|xi − yi|.

As a consequence of Lemma 7.10, E` is a Morse-Bott function whose critical man-ifolds are products C1 × · · · × C` of critical manifolds of E, so each Ci is either abinormal chord or the corresponding diagonal. Note that the symmetric group S`acts on (K ×K)` preserving E` as well as the product metric.

For a > 0 we denote by Ma ⊂ (K ×K)` the collection of tuples c = (c1, . . . , c`) ofbinormal chords of total length L(c) = a, and byW a the disjoint union of the unstablemanifolds of points in Ma under the flow of −∇E` (here Ma and thus W a may beempty). Let φT : (K ×K)` → (K ×K)` be the time-T map of the flow of −∇E`.

Lemma 7.11. — For a generic knot K ⊂ R3 and each a > 0 there exist εa > 0 andTa > 0 with the following property. For each ε < εa, T > Ta and ` ∈ N we have

φT (L` 6 a+ ε) ⊂ L` 6 a− ε ∪ V a,

where V a is a tubular neighborhood of W a ∩ L` > a − ε in a − ε 6 L` 6 a + ε.Moreover, tuples of Q-strings in V a do not intersect the knot K in their interior.

Proof. — Note that on K × K the length and energy are related by L =√

2E, sothey have the same critical points and L is strictly decreasing under the flow of −∇Eoutside the critical points. Since the flow of −∇E` is the product of the flows of E ineach factor, the same relation holds for any ` ∈ N: L` and E` have the same criticalpoints and L` is strictly decreasing under the flow of −∇E` outside the critical points.

Next recall from above that E` is a Morse-Bott function. In particular, the set ofcritical values of E`, and thus also of L`, is discrete. Given a ∈ R, we pick εa > 0

such that a is the only critical value of L` in the interval [a− εa, a+ εa]. (Since onlyfinitely many binormal chords can appear in tuples of critical points of total length a,the constant εa can be chosen independently of `.) For ε < εa, the familiar argumentfrom Morse theory shows that φT (L` 6 a + ε) ⊂ L` 6 a − ε ∪ V aε,T , where V aε,T

J.É.P. — M., 2017, tome 4


for large T are tubular neighborhoods of W a ∩ L` > a− ε in a− ε 6 L` 6 a+ εthat shrink to W a ∩ L` > a− ε as T →∞.

For the last statement recall that, for a generic knot K, binormal chords do notmeet K in their interior. So for each ` ∈ N there exists a neighborhood Ua of Ma

in (K ×K)` such that tuples of Q-strings in Ua do not intersect K in their interior.We pick Ta large enough and εa small enough so that V aεa,Ta is contained in Ua. Bythe argument as in the previous paragraph, the constants εa and Ta can be chosenindependently of ` and the lemma is proved.

7.5. Shortening linear Q-strings. — We will need some homological algebra. Sup-pose we have the following algebraic situation:

– a chain complex (C , D = ∂ + δ) satisfying the relations

∂2 = δ2 = ∂δ + δ∂ = 0, and

– a chain map f : (C , ∂) → (C , ∂) and a chain homotopy H : (C , ∂) → (C , ∂)

satisfying

(7.4) ∂H +H∂ = f − 1l,

such that for every c ∈ C there exists a positive integer S(c) with

(7.5) (δH)S(c)(c) = 0.

In our applications below, we will have δ = δQ + δN , and the equation δ2 = 0 willfollow from

δ2Q = δ2

N = [δQ, δN ] = 0,

which is part of the statement that D2 = 0 in our chain complex. Here, as usual, wedenote the graded commutator of two maps A,B by

[A,B] := AB − (−1)|A||B|BA.

Set H0 := H and f0 := f , and more generally for d > 1 define the maps

(7.6) Hd := H(δH)d, fd :=

d∑i=0

(Hδ)if(δH)d−i.

It is also convenient to set H−1 = 0. Note that the maps fd satisfy the recursionrelation fd+1 = fdδH +Hdδf .

Lemma 7.12. — For each d > 1 we have

(7.7) [∂,Hd] + [δ,Hd−1] = fd.

J.É.P. — M., 2017, tome 4


Proof. — We prove this by induction on d. The case d = 1 is an immediate conse-quence of (7.4) and [δ, ∂] = 0. For the induction step we observe that

[∂,Hd+1] = ∂HdδH +HdδH∂

= [∂,Hd]δH −Hd∂δH −Hdδ∂H +Hdδf −Hdδ

= fdδH − [δ,Hd−1]δH +Hdδf −Hdδ

= fd+1 − δHd −Hd−1δ2H −Hdδ

= fd+1 − [δ,Hd].

Here in the second equality we have used (7.4), in the third equality the inductionhypothesis and [δ, ∂] = 0, in the fourth equality the recursion relation above, and inthe fifth equality we have used δ2 = 0.

In view of equation (7.5), for each c ∈ C we have Hdc = 0 and fdc = 0 for

d > S(c) + maxS(fc), S(fδHc), . . . , S(f(δH)S(c)−1c)

.

So the sums

(7.8) H :=

∞∑d=0

Hd, F :=

∞∑d=0

fd

are finite on every c ∈ C . Summing up equation (7.7) for d = 1, . . . , e and usingequation (7.4), we obtain

[∂,He] + [D,H0 + · · ·+He−1] = f0 + · · ·+ fe − 1l

for all e, and hence[D,H] = F− 1l.

This concludes the homological algebra discussion.We now apply this construction to the space Σlin of broken strings with linear

Q-strings as follows. We fix a large time T > 0 and consider a generic i-chain β

in Σlin, for i = 0, 1. Moving the Q-strings in β by the flow of −∇E for times t ∈ [0, T ]

we obtain an (i + 1)-chain in (K ×K)`. We make this an (i + 1)-chain HTβ in Σlin

by dragging along the N -strings without creating new intersections with the knot.In the case i = 1, we moreover grow new N -spikes starting from the finitely manypoints Zβ where some Q-string becomes tangent to the knot at one end point, asshown in Figure 7.3. We define fTβ as the boundary component of HTβ at time T .

Remark 7.13. — Technically, we should be careful to arrange that H maps genericchains to generic chains. This is easy for 0-chains, but some care should be taken for1-chains, especially near the points Zβ where some Q-string becomes tangent to Kat one of its end points.

Proposition 7.14. — For a generic knot K, the operations defined above yield fori = 0, 1 maps

fT : Ci(Σlin) −→ Ci(Σlin), HT : Ci(Σlin) −→ Ci+1(Σlin)

satisfying conditions (7.4) and (7.5).

J.É.P. — M., 2017, tome 4


Proof. — Standard transversality arguments show that fT and HT map genericchains to generic chains, provided that we impose suitable genericity conditions ongeneric chains with respect to linear strings. Now condition (7.4) is clear by construc-tion.

For condition (7.5), we use Lemma 7.10(c). It implies that there exists a neighbor-hood U ⊂ K ×K of the finitely many 2-gons ∂SQ that are tangent to K at one endpoint and an ε > 0 with the following property: Each 2-gon in U ∩ SQ decreases inlength by at least ε under the flow of −∇E before it meets SQ again, and the sameholds for the longer 2-gon resulting from splitting it at its intersection with the knot.On the other hand, if a 2-gon in SQrU is split at its intersection with the knot, thenboth pieces are shorter by at least some fixed amount δ > 0. Hence each applicationof HT δQ decreases the total length of Q-strings by at least min(ε, δ), and since L(β)

is finite this can happen only finitely many times.

Applying definition (7.8) to the maps fT and HT , we obtain for i = 0, 1 lengthdecreasing maps

FT : Ci(Σlin) −→ Ci(Σlin), HT : Ci(Σlin) −→ Ci+1(Σlin)

satisfying

(7.9) DHT0 = FT0 − 1l, HT0 D +DHT1 = FT1 − 1l

We now use these maps to compute the homology of (Ci(Σlin), D) in small lengthintervals. For a ∈ R and i = 0, 1 we denote by A a

i the free Z-module generated bywords γ1c1 · · · γ`c`γ`+1, ` > 0, where c1, . . . , c` are binormal chords of total length aand of total index i, and the γj are homotopy classes of paths in ∂N connecting the cjto broken strings and not intersecting K in their interior. We define linear maps

Θ : A ai −→ H

[a−ε,a+ε)i (Σlin, D)

as follows. For i = 0, Θ sends γ1c1 · · · γ`c`γ`+1 to the homology class of the brokenstring γ1c1 · · · γ`c`γ`+1, where γj are representatives of the classes γj . For i = 1,consider a word γ1c1 · · · γ`c`γ`+1 with exactly one binormal chord ck of index 1 andall others of index 0. Then Θ sends this word to the homology class of the 1-chainγ1c1 · · · ck · · · γ`c`γ`+1, where γj are representatives of the classes γj and ck is theunstable manifold of ck in (K ×K)∩L > a− ε, viewed as a 1-chain by fixing someparametrization.

Corollary 7.15. — For a ∈ R let εa be the constant from Lemma 7.11. Then foreach ε < εa the map Θ : A a

i → H[a−ε,a+ε)i (Σlin, D) is an isomorphism for i = 0 and

surjective for i = 1.

Proof. — We first consider the case i = 1. Fix ε < εa and T > Ta, where εa, Ta arethe constants from Lemma 7.11. Consider a relative 1-cycle β ∈ C [a−ε,a+ε)

1 (Σlin). Inview of (7.9), β is homologous to FTβ. Recall from its definition in (7.6) and (7.8)that each tuple of Q-strings appearing in FTβ is obtained by flowing some tuple of

J.É.P. — M., 2017, tome 4


Q-strings for time T (and maybe applying HδQ several times to the resulting tuple).Now we distinguish two cases.

Case 1. — a is not the length of a word of binormal chords. Then in Lemma 7.11 theset V a is empty and it follows that all tuples of Q-strings in FTβ have length at mosta− ε. This shows that H [a−ε,a+ε)

1 (Σlin) = 0 and the map Θ is an isomorphism.

Case 2. — a is the length of a word of binormal chords. For simplicity, let us assumethat up to permutation there is only one word w of length a (the general case differsjust in notation). By Lemma 7.11, FTβ is a finite sum β′1 + β′2 + · · · of relative 1-cycles β′` in tubular neighborhoods V a of the unstable manifoldsW a∩L` > a−ε ofcritical `-tuples of length a. Recall that critical `-tuples consist of binormal chords andQ-spikes (corresponding to constant 2-gons). Using the operation δN , we can replaceQ-spikes by differences of N -strings to obtain a relative 1-cycle β′′ in V a homologousto FTβ which contains no Q-spikes. So each 1-simplex β′′j in β′′ is a relative 1-chainwhose Q-strings lie in the tubular neighborhood Vj of the unstable manifold of somepermutation wj of w. Then the N -strings in β′′ do not intersect the knot in theirinterior, and by Lemma 7.11 neither do the Q-strings. Thus each β′′j is a relative cyclein Vj with respect to the singular boundary ∂. We distinguish three subcases.

(i) If the total degree of the word w is bigger than 1, then its stable manifoldfor the flow of −∇E has codimension bigger than 1. So, after a small perturbation,each β′′j will avoid the stable manifold of wj and will therefore have length at mosta− ε for sufficiently large T . This shows that, as in Case 1, both groups vanish and Θ

is an isomorphism.(ii) If the degree of the word w is 0, then its unstable manifold is a point and thus

each Vj is contractible relative to L 6 a− ε. It follows that each relative cycle β′′j is∂-exact, and since no δQ and δN occurs also D-exact. Again we see that both groupsvanish and Θ is an isomorphism.

(iii) If the degree of the word w is 1, then each Vj deformation retracts relative toL 6 a− ε onto the 1-dimensional unstable manifold wj of wj . It follows that eachrelative cycle β′′j is ∂-homologous, and since no δQ and δN occurs also D-homologous,to a multiple of the 1-chain of Q-strings wj connected by suitable N -strings. Bydefinition of Θ, this shows that the D-homology class [β′′] = [β] lies in the imageof Θ. So Θ is surjective, which concludes the case i = 1.

In the case i = 0, the proof of surjectivity is analogous but simpler than in the casei = 1. For injectivity one considers FTβ for a 1-chain β in Σlin with Dβ = α for agiven 0-chain α and argues similarly. Note that this last step does not work to proveinjectivity for i = 1 because it would require considering FTβ for a 2-chain β, whichwe have not defined (although this should of course be possible).

7.6. Proof of the isomorphism. — Let Φ : (C∗(R), ∂Λ) → (C∗(Σ), D) be the chainmap constructed in the previous section. We now use the fact (Corollary 6.15) thatthe map Φ preserves the length filtrations. Thus for a < b < c we have the commuting

J.É.P. — M., 2017, tome 4


diagram with exact rows of length filtered homology groups

H[b,c)1 (R) −−−−→ H

[a,b)0 (R) −−−−→ H

[a,c)0 (R) −−−−→ H

[b,c)0 (R) −−−−→ 0yΦ∗

yΦ∗

yΦ∗

yΦ∗

yH

[b,c)1 (Σ) −−−−→ H

[a,b)0 (Σ) −−−−→ H

[a,c)0 (Σ) −−−−→ H

[b,c)0 (Σ) −−−−→ 0 .

The main result of this section asserts that Φ∗ is an isomorphism (resp. surjective)for sufficiently small action intervals:

Proposition 7.16. — For each a ∈ R there exists an εa > 0 such that for each ε < εathe map

Φ∗ : H[a−ε,a+ε)0 (R) −→ H

[a−ε,a+ε)0 (Σ)

is an isomorphism and the map

Φ∗ : H[a−ε,a+ε)1 (R) −→ H

[a−ε,a+ε)1 (Σ)

is surjective.

This proposition implies Theorem 1.2 as follows. SinceH0(R) = limR→∞H[0,R)0 (R)

and H0(Σ) = limR→∞H[0,R)0 (Σ), it suffices to show that

Φ∗ : H[0,R)0 (R) −→ H

[0,R)0 (Σ)

is an isomorphism for each R > 0. Now the compact interval [0, R] is covered byfinitely many of the open intervals (a − εa, a + εa), with a ∈ [0, R] and εa as inProposition 7.16. Thus, according to Proposition 7.16, there exists a partition 0 =

r0 < r1 < · · · < rN = R such that the maps

Φ∗ : H[ri−1,ri)0 (R) −→ H

[ri−1,ri)0 (Σ)

are isomorphisms and

Φ∗ : H[ri−1,ri)1 (R) −→ H

[ri−1,ri)1 (Σ)

are surjective for all i = 1, . . . , N . To prove by induction that

Φ∗ : H[0,ri)0 (R) −→ H

[0,ri)0 (Σ)

is an isomorphism for each i = 1, . . . , N , consider the commuting diagram above witha = 0, b = ri−1 and c = ri. By induction hypothesis for i − 1 the second, fourthand fifth vertical maps are isomorphisms and the first one is surjective, so by the fivelemma the third vertical map is an isomorphism as well. This proves the inductivestep and hence Theorem 1.2.

Proof of Proposition 7.16. — Let us denote the maps provided by Proposition 7.1 byFpli ,H

pli and the maps in Proposition 7.6 by Flin

i ,Hlini , i = 0, 1. A short computation

shows that the maps

Fi := Flini F

pli : Ci(Σ) −→ Ci(Σlin),

Hi := Hpli + ipl Hlin

i Fpli : Ci(Σ) −→ Ci+1(Σ)

for i = 0, 1 satisfy with the map i := ipl ilin : C∗(Σlin) → C∗(Σ):

J.É.P. — M., 2017, tome 4


(i) F0i = 1l and DH0 = iF0 − 1l;(ii) F1i = 1l and H0D +DH1 = iF1 − 1l;(iii) F0, H0, F1 and H1 are (not necessarily strictly) length-decreasing.

Conditions (i) and (ii) imply DF1 = F0D and iF1D = D(1l + H1D), and therefore

F0(imD) ⊂ imD, F1(kerD) ⊂ kerD, F1(imD) ⊂ i−1(imD).

Hence the Fi define chain maps between the chain complexes (where the left horizontalmaps are the obvious inclusions)

imD −−−−→ C1(Σ)D−−−−→ C0(Σ)yF1

yF1

yF0

i−1(imD) −−−−→ C1(Σlin)Dlin

−−−−→ C0(Σlin) .

Note that the upper complex computes the homology groups H0(Σ) and H1(Σ), whilethe lower complex has homology groups H0(Σlin) and

H1(Σlin) := kerDlin/i−1(imD).

Conditions (i) and (ii) show that F0,F1 induce isomorphisms between these homologygroups (with inverses i∗), and in view of condition (iii) the same holds for lengthfiltered homology groups. Setting

Ψ := Fi Φ : (Ci(R), ∂Λ) −→ (Ci(Σlin), D), i = 0, 1,

it therefore suffices to prove: For each a ∈ R there exists an εa > 0 such that for eachε < εa the map

Ψ∗ : H[a−ε,a+ε)0 (R) −→ H

[a−ε,a+ε)0 (Σlin)

is an isomorphism and the map

Ψ∗ : H[a−ε,a+ε)1 (R) −→ H


is surjective.We take for εa the constant from Lemma 7.11 and consider ε < εa. Then we have

canonical isomorphisms

Γ : H[a−ε,a+ε)i (R) ∼= A a

i , i = 0, 1

to the groups A ai introduced in the previous subsection. Recall the maps Θ : A a

i →H

[a−ε,a+ε)i (Σlin, D) from Corollary 7.15 which are an isomorphism for i = 0 and

surjective for i = 1.We consider first the case i = 0. By Proposition 6.14, for a binormal chord c of

index 0 and length a the moduli space of holomorphic disks with positive puncture cand switching boundary conditions contains one component corresponding to the half-strip over c, and on all other components the Q-strings in the boundary have totallength less than a−ε. This shows that the map Ψ∗ : H

[a−ε,a+ε)0 (R)→ H


agrees with Θ Γ and is therefore an isomorphism.

J.É.P. — M., 2017, tome 4


For i = 1 we have a diagram

H[a−ε,a+ε)1 (R)

Ψ∗−−−−→ H[a−ε,a+ε)1 (Σlin)

∼=yΓ

xΠ

A a1

Θ−−−−→ H[a−ε,a+ε)1 (Σlin),

where Π : H1(Σlin) = kerDlin/ imDlin → kerDlin/i−1(imD) = H1(Σlin) is the canon-ical projection. Since Π and Θ are surjective, surjectivity of Ψ∗ follows once we showthat the diagram commutes.

To see this, consider a word w = b1 · · · bkc of binormal chords of indices |bi| = 0

and |c| = 1 and total length a. The 1-dimensional moduli space of holomorphic stripswith positive puncture asymptotic to c and one boundary component on the zerosection contains a unique component Mc passing through the trivial strip over c. ByProposition 6.14, for each other element in Mc the boundary on the zero section haslength strictly less than L(c). So, for ε sufficiently small, the moduli space representsa generator of the local first homology at c. Since on all other components of themoduli space the Q-strings in the boundary have total length less than a − ε, theproduct of Mc with the half-strips over the bj gives Φ(w) ∈ C [a−ε,a+ε)

1 (Σ). Its imageΨ(w) = F1 Φ(w) ∈ C

[a−ε,a+ε)1 (Σlin) is obtained from Φ(w) by shortening the Q-

strings to linear ones. Since the tuples of Q-strings in Φ(w) were either C1-closeto w (depending on ε) or had total length less that a − ε, the same holds for Ψ(w).Hence Ψ(w) is homologous (with respect to ∂, and therefore with respect to D) inC

[a−ε,a+ε)1 (Σlin) to the unstable manifold of w in Σlin, which by definition equals

Π Θ Γ(w).In the previous argument we have ignored the N -strings, always connecting the

ends of Q-strings to the base point by capping paths. More generally, a generator ofH

[a−ε,a+ε)1 (R) ∼= A a

1 is given by a word γ1c1 · · · γ`c`γ`+1, where the cj are binormalchords with one of them of index 1 and all others of index 1, and the γj are homo-topy classes of N -strings connecting the end points and not intersecting K in theinterior. Now we apply the same arguments as above to the Q-strings, dragging alongthe N -strings, to prove commutativity of the diagram. This concludes the proof ofProposition 7.16, and thus of Theorem 1.2.

8. Properties of holomorphic disks

In this section we begin our analysis of the holomorphic disks involved in thedefinition of the chain map from Legendrian contact homology to string homology.For the remainder of the paper, we consider the following setup:

– Q is a real analytic Riemannian 3-manifold without closed geodesics and convexat infinity (the main example being Q = R3 with the flat metric);

– K ⊂ Q is a real analytic knot with nondegenerate binormal chords;

J.É.P. — M., 2017, tome 4


– LK ⊂ T ∗Q is the conormal bundle, Q ⊂ T ∗Q is the 0-section, and

L = LK ∪Q

is the singular Lagrangian with clean intersection LK ∩Q = K.The reader will notice that much of the discussion naturally extends to higher

dimensional manifolds Q and submanifolds K ⊂ Q.

8.1. Almost complex structures. — Consider the subsets

S∗Q = (q, p)∣∣ |p| = 1 ⊂ D∗Q = (q, p)

∣∣ |p| 6 1 ⊂ T ∗Q

of the cotangent bundle. The canonical isomorphism

R× S∗Q −→ T ∗QrQ,(s, (q, p)

)7−→ (q, esp)

intertwines the R-actions given by translation resp. rescaling. Let λ = p dq be thecanonical Liouville form on T ∗Q with Liouville vector field p∂p. Its restriction λ1 toS∗Q is a contact form with contact structure ξ = kerλ1 and Reeb vector field R.We denote the R-invariant extensions of λ1, ξ, R to T ∗QrQ by the same letters. Ingeodesic normal coordinates qi and dual coordinates pi they are given by

λ1 =p dq

|p|, R =

∑i

pi∂

∂qi, ξ(q,p) = kerλ1 ∩ ker(p dp) = span

R, p

∂

∂p

⊥dλ1.

Around each Reeb chord c : [0, T ] → S∗Q with end points on ΛK = LK ∩ S∗Q wepick a neighborhood U × (−ε, T + ε) ⊂ S∗Q, where U is a neighborhood of the originin C2, with the following properties:

– the Reeb chord c corresponds to 0 × [0, T ];– the Reeb vector field R is parallel to ∂t, where t is the coordinate on (−ε, T + ε)

and the contact planes project isomorphically onto U along R;– along 0× (−ε, T + ε) the contact planes agree with C2×0 and the form dλ1

with ωst = dx1 ∧ dy1 + dx2 ∧ dy2;– the Legendrian ΛK intersects U×(−ε, T+ε) in two linear subspaces contained in

U×0 and U×T, respectively, whose projections to U are transversely intersectingLagrangian subspaces of (C2, ωst).

Definition 8.1. — An almost complex structure J on T ∗Q is called admissible if ithas the following properties.

(i) J is everywhere compatible with the symplectic form dp∧ dq. Moreover, Q ad-mits an exhaustion Q1 ⊂ Q2 ⊂ · · · by compact sets with smooth boundary such thatthe pullbacks π−1(∂Qi) under the projection π : T ∗Q → Q are J-convex hypersur-faces.

(ii) Outside D∗Q, J agrees with an R-invariant almost complex structure J1 on thesymplectization that takes the Liouville field p∂p to the Reeb vector field R, restrictsto a complex structure on the contact distribution ξ, and is compatible with thesymplectic form dλ1 on ξ.

J.É.P. — M., 2017, tome 4


(iii) Outside the zero section, J preserves the subspace spanp∂p, R as well as ξand is compatible with the symplectic form dλ1 on ξ. Along the zero section, J agreeswith the canonical structure ∂/∂pi 7→ ∂/∂qi.

(iv) J is integrable near K such that Q and K are real analytic.(v) On each neighborhood U × (−ε, T + ε) around a Reeb chord as above, the

restriction of J1 to the contact planes is the pullback of the standard complex structureon U ⊂ C2 under the projection.

Remark 8.2. — Conditions (i) and (ii) are standard conditions for studying holo-morphic curves in T ∗Q and its symplectization R× S∗Q. Condition (iii) ensures thecrucial length estimate for holomorphic curves in the next subsection. Condition (iv)is needed for the Finiteness Theorem 6.5 to hold. Condition (v) is added to facilitateour study of spaces of holomorphic disks and is convenient for fixing gauge whenfinding smooth structures on moduli spaces; it can probably be removed with a moreinvolved analysis of asymptotics.

Remark 8.3. — Note that an admissible almost complex structure remains so underarbitrary deformations satisfying (ii) that are supported outside D∗Q and away fromthe Reeb chords. This gives us enough freedom to achieve transversality within theclass of admissible structures in Section 9.

The Riemannian metric on Q induces a canonical almost complex structure Jst onT ∗Q which in geodesic normal coordinates qi at a point q and dual coordinates pi isgiven by

Jst

(∂

∂qi

)= − ∂

∂pi, Jst

(∂

∂pi

)=

∂

∂qi.

More generally, for a positive smooth function ρ : [0,∞)→ (0,∞) we define an almostcomplex structure Jρ by

Jρ

(∂

∂qi

)= −ρ(|p|) ∂

∂pi, Jρ

(∂

∂pi

)= ρ(|p|)−1 ∂

∂qi.

If ρ(r) = r for large r, then it is easy to check that Jρ satisfies the first part ofcondition (i) as well as conditions (ii) and (iii) in Definition 8.1. If the metric is flat(i.e., Q is R3 or a quotient of R3 by a lattice), then Jst is integrable and Jρ alsosatisfies the second part of (i) (choosing Qi to be round balls) and condition (iv).Condition (v) can then be arranged by deforming Jρ near infinity within the class ofalmost complex structures satisfying (ii). So we have shown the following.

Lemma 8.4. — For Q = R3 with the Euclidean metric there exist admissible almostcomplex structures in the sense of Definition 8.1.

Remark 8.5. — In fact, the almost complex structure Jst induced by the metric isintegrable if and only if the metric is flat (this observation is due to M. Grüneberg,unpublished). So the preceding proof of Lemma 8.4 does not carry over to generalmanifolds Q (although the conclusion should still hold).

J.É.P. — M., 2017, tome 4


The next result provides nice holomorphic coordinates near K ⊂ T ∗Q.

Lemma 8.6. — Suppose that J satisfies condition (iv) in Definition 8.1. Then forδ > 0 small enough there exists a holomorphic embedding from S1 × (−δ, δ) × B4

δ ,where B4

δ ⊂ C2 is the ball of radius δ, with its standard complex structure onto aneighborhood of K in T ∗Q with complex structure J with the following properties:

– S1 × 0 × 0 maps onto K;– S1 × 0 × (R2 ∩B4

δ ) maps to Q;– S1 × 0 × (iR2 ∩B4

δ ) maps to LK .

Alternatively, we can arrange the last two properties with the roles of Q and LKinterchanged.

Proof. — This is proved in more generality in [5, Rem. 3.2]; for convenience we repeatthe proof in the situation at hand. Consider the real analytic embedding γ : S1 → Q

representing K. Pick a real analytic vector field v on Q which is nowhere tangent to Kalong K. Let v1 be the unit vector field along K in the direction of the componentof v perpendicular to γ. Then v1 is a real analytic vector field along K. Let v2 = γ×v1

be the unit vector field along K which is perpendicular to both γ and v1 and whichis such that (γ, v1, v2) is a positively oriented basis of TQ. Consider S1 × D2 withcoordinates (s, σ1, σ2), s ∈ R/Z, σj ∈ R. Since K is an embedding there exists ρ > 0

such that

(8.1) φ(s, σ1, σ2) = γ(s) + σ1v1(s) + σ2v2(s)

is an embedding for σ21+σ

22<ρ. Note that the embedding is real analytic. Equip S1×D2

with the flat metric and consider the induced complex structure on T ∗(S1×D2).The real analyticity of φ in (8.1) implies that it extends to holomorphic embedding Φ

from a neighborhood of S1×D2 in T ∗(S1×D2) to a neighborhood of K in T ∗Q (herewe use integrability of J near K). In fact, locally Φ is obtained by replacing the realvariables (s, σ1, σ2) in the power series corresponding in the right hand side of (8.1)by their complexifications (s+ it, σ1 + iτ1, σ2 + iτ2). This proves the first assertion ofthe lemma. The alternative assertion follows from this one by precomposing Φ withmultiplication by i on B4

δ .

Remark 8.7. — The coordinate system gives a framing ofK determined by the normalvector field v. By real analytic approximation we can take v to represent any class offramings.

8.2. Length estimates. — In this subsection we show that the chain map Φ respectsthe length filtrations. This was shown in [6] for the absolute case, i.e., without theadditional boundary condition LK , and the arguments carry over immediately to therelative case. For completeness, we provide the proof in this subsection and we keepthe level of generality of [6], which is slightly more than what we use in this paper.

J.É.P. — M., 2017, tome 4


For preparation, consider a smooth function τ : [0,∞) → [0,∞) with τ ′(s) > 0

everywhere and τ(s) = 0 near s = 0. Then

λτ :=τ(|p|)p dq|p|

defines a smooth 1-form on T ∗Q.

Lemma 8.8. — Let J be an admissible almost complex structure on T ∗Q and τ a func-tion as above. Then for all v ∈ T(q,p)T

∗Q we have

dλτ (v, Jv) > 0.

At points where τ(|p|) > 0 and τ ′(|p|) > 0 equality holds only for v = 0, whereasat points where τ(|p|) > 0 and τ ′(|p|) = 0 equality holds if and only if v is a linearcombination of the Liouville field p ∂p and the Reeb vector field R = p ∂q.

Proof. — By condition (iii) in Definition 8.1, J preserves the splittingT (T ∗Q) = spanp ∂p, R ⊕ ξ

and is compatible with dλ1 on ξ. Let us denote by π1 : T (T ∗Q) → spanp ∂p, Rand π2 : T (T ∗Q) → ξ the projections onto the direct summands. Since ker(dλ1) =

spanp ∂p, R, for v ∈ T(q,p)T∗Q we conclude

dλ1(v, Jv) = dλ1(π2v, Jπ2v) > 0,

with equality iff v ∈ spanp ∂p, R. Next, we consider

dλτ = τ(|p|)dλ1 +τ ′(|p|)|p|

p dp ∧ λ1.

Since the form p dp ∧ λ1 vanishes on ξ and is positive on spanp ∂p, R, we conclude

dλτ (v, Jv) = τ(|p|)dλ1(π2v, Jπ2v) +τ ′(|p|)|p|

p dp ∧ λ1(π1v, Jπ1v) > 0,

with equality iff both summands vanish. From this the lemma follows.

Let now J be an admissible almost complex structure on T ∗Q and

u : (Σ, ∂Σ) −→ (T ∗Q,Q ∪ LK)

be a J-holomorphic curve with finitely many positive boundary punctures asymptoticto Reeb chords a1, . . . , as and with switching boundary conditions on Q ∪ LK . Letσ1, . . . , σk be the boundary segments on Q. Recall that L(σi) denotes the Riemannianlength of σi and L(aj) =

∫ajλ1 denotes the action of the Reeb chord aj , which agrees

with the length of the corresponding binormal chord.

Proposition 8.9. — With notation as above we havek∑i=1

L(σi) 6s∑j=1

L(aj),

and equality holds if and only if u is a branched covering of a half-strip over a binormalchord.

J.É.P. — M., 2017, tome 4


Proof. — The idea of the proof is straightforward: integrate u∗dλ1 over Σ and applyStokes’ theorem. However, some care is required to make this rigorous because the1-form λ1 is singular along the zero section.

Fix a small δ > 0. For i = 1, . . . , s pick biholomorphic maps φi : [0, δ] × [0, 1] →Ni ⊂ Σ onto neighborhoods Ni in Σ of the ith boundary segment mapped to Q, sothat φi(0, t) is a parametrization of the ith boundary segment. We choose δ so smallthat Ni ∩Nj = ∅ if i 6= j and u φi(δ, ·) does not hit the zero section (the latter ispossible because otherwise by unique continuation u would be entirely contained inthe zero section, which it is not by assumption). For fixed i we denote the inducedparametrization of σi by q(t) := u φi(t) ∈ Q, so we can write

u φi(s, t) = (q(t) + v(s, t), sq(t) + w(s, t))

with v(0, t) = 0 = w(0, t), and therefore ∂v/∂t(0, t) = 0 = ∂w/∂t(0, t). The hypothesisthat J is standard near the zero section (condition (iii) in Definition 8.1) implies that∂v/∂s(0, t) = 0 = ∂w/∂s(0, t). Denoting vδ = v(δ, ·) and wδ = w(δ, ·) we compute

(u φi)∗λ1|s=δ =〈δq + wδ, q + vδ〉|δq + wδ|

dt

=〈q + wδ/δ, q + vδ〉|q + wδ/δ|

dt

=(|q|+O(δ)

)dt,

where in the last line we have used that vδ = O(δ) and wδ = O(δ2).Pick ε > 0 smaller than the minimal norm of the p-components of uφi(δ, ·) for all

i. Pick a function τ : [0,∞)→ [0, 1] with τ ′ > 0, τ(s) = 0 near s = 0, and τ(s) = 1 fors > ε. By Lemma 8.8, the form λτ = (τ(|p|)/|p|) pdq on T ∗Q satisfies u∗(dλτ ) > 0.Note that λτ agrees with λ1 = (p/|p|) dq on the subset |p| > ε ⊂ T ∗Q, so thepreceding computation yields∫

s=δ(u φi)∗λτ =

∫s=δ

(|q|+O(δ)

)dt = L(σi) +O(δ)

for all i. Next, consider polar coordinates (r, ϕ) around 0 in the upper half plane H+

near the jth positive puncture. Then the asymptotic behavior of u near the puncturesyields ∫

r=δ∩H+

u∗λτ = L(aj) +O(δ).

Now let Σδ ⊂ Σ be the surface obtained by removing the neighborhoods r 6 δ∩H+

around the positive punctures and the neighborhoods Ni of the boundary segmentsmapped to Q, see Figure 8.1.

The boundary of Σδ consists of the arcs r = δ∩H+ around the positive punctures,the arcs φi(s = δ) near the boundary segments mapped to Q (negatively oriented),and the remaining parts of ∂Σ mapped to LK . Since λτ vanishes on LK , the latterboundary parts do not contribute to its integral and Stokes’ theorem combined with

J.É.P. — M., 2017, tome 4


LK LK

LK

LK

LK

Q

Q

Q Σ

Σδ

N1

N2

N3

Figure 8.1. The domain Σδ is obtained from Σ by removing smallneighborhoods of the boundary arcs mapping to Q and of the positivepunctures. The punctures are denoted by x, and switches are denotedby dots.

the preceding observations yields

0 6∫

Σδ

u∗dλτ =

∫∂Σδ

u∗λτ =

s∑j=1

L(aj)−k∑i=1

L(σi) +O(δ).

Taking δ → 0 this proves the inequality in Proposition 8.9. Equality holds iff u∗dλτvanishes identically, which by Lemma 8.8 is the case iff u is everywhere tangent tospanp ∂p, R. In view of the asymptotics at the positive punctures, this is the caseprecisely for a half-strip over a binormal chord.

8.3. Holomorphic half-strips. — We consider the half-strip R+ × [0, 1] with co-ordinates (s, t) and its standard complex structure. Let J be an admissible almostcomplex structure on T ∗Q and J1 the associated structure on R×S∗Q. A holomorphichalf-strip in R× S∗Q is a holomorphic map

u : R+ × [0, 1] −→ (R× S∗Q, J1)

mapping the boundary segments R× 0 and R× 1 to R× ΛK . Similarly, a holo-morphic half-strip in T ∗Q is a holomorphic map

u : R+ × [0, 1] −→ (T ∗Q, J)

mapping the boundary to L = LK ∪ Q. We write the components of a map u intoR× S∗Q (or into T ∗QrD∗Q ∼= R+ × S∗Q) as

u = (a, f).

Recall from [3] (see also [5]) that to any smooth map u from a surface to R× S∗Q orT ∗Q we can associate its Hofer energy E(u). It is defined as the sum of two terms,the ω-energy and the λ-energy, whose precise definition will not be needed here. Thefollowing result follows from [10, Lem.B.1], see also [3, Prop. 6.2], in combination withwell-known results in Lagrangian Floer theory, see e.g. [21].

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Proposition 8.10. — For each holomorphic half-strip u in R× S∗Q or T ∗Q of finiteHofer energy exactly one of the following holds:

– There exists a Reeb chord c : [0, T ]→ S∗Q and a constant a0 ∈ R such that

a(s, t)− Ts− a0 −→ 0, f(s, t) −→ c(Tt)

uniformly in t as s→∞. We say that the map has a positive puncture at c.– There exists a Reeb chord c : [0, T ]→ S∗Q and a constant a0 ∈ R such that

a(s, t) + Ts− a0 −→ 0, f(s, t) −→ c(−Tt)

uniformly in t as s→∞. We say that the map has a negative puncture at c.– There exists a point x0 on R× ΛK (resp. L) such that

u(s, t) −→ x0

uniformly in t as s→∞. In this case u χ−1, where χ : R+× [0, 1]→ D+ is the mapfrom (6.3), extends to a holomorphic map on the half-disk mapping the boundary toR×ΛK (resp. L). If x0 /∈ K then we say that u has a removable puncture at x0, andif x0 ∈ K then we say that u has a Lagrangian intersection puncture at x0. (Theseare the standard situations in ordinary Lagrangian intersection Floer homology.)

Because of our choice of almost complex structure we can say more about the localforms of the maps as follows.

Consider first a Reeb chord puncture where the map approaches a Reeb chord c.Let U × (−ε, T + ε) be the neighborhood of c as in Definition 8.1 (v) and note thatthe holomorphic half-strip is uniquely determined by the local projection to U ⊂ C2

where the complex structure is standard. By a complex linear change of coordinateson C2 we can arrange that the two branches of the Legendrian ΛK through the endpoints of c project to R2 and to the subspace spanned by the vectors (eiθ1 , 0) and(0, eiθ2), for some angles θ1, θ2. The C2-component v of the map u then has a Fourierexpansion

(8.2) v(z) =∑n>0

(c1;ne

−(θ1+n)z, c2;ne−(θ2+n)z

),

where cj;n are real numbers. We call the smallest n such that (c1;n, c2;n) 6= 0 the orderof convergence to the Reeb chord c.

We have similar expansions near the Lagrangian intersection punctures. Lemma 8.6gives holomorphic coordinates (z0, z1) = (x0+iy0, x1+iy1) in C×C2 around any pointq0 ∈ K such that the Lagrangian submanifoldQ ⊂ T ∗Q corresponds to y0 = y1 = 0,the Lagrangian submanifold LK corresponds to y0 = x1 = 0, and the almost com-plex structure J corresponds to the standard complex structure i on C3. Consider aholomorphic map u : [0,∞)× [0, 1]→ T ∗Q such that u(z)→ q ∈ K as z →∞ where qlies in a small neighborhood of q0 in K. We write u in the local coordinates describedabove as v = (v0, v1). Now Remark 4.2 yields the following Fourier expansions for v.

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If v([0,∞)× 0) ⊂ Q and v([0,∞)× 1) ⊂ LK then

(8.3) v(z) =

(∑m>0

c0,me−mπz,

∑n+1/2>0

c1;n+1/2 e−(n+1/2)πz

),

where c0;m ∈ R for all m ∈ Z>0 and where c1;n+1/2 ∈ R2 for all n ∈ Z>0, in aneighborhood of ∞. If v([0,∞)× 0) ⊂ LK and v([0,∞)× 1) ⊂ Q then

(8.4) v(z) =

(∑m>0

c0,me−mπz, i

∑n+1/2>0

c1;n+1/2 e−(n+1/2)πz

),

where notation is as in (8.3). If v([0,∞)× 0) ⊂ Q and v([0,∞)× 1) ⊂ Q then

(8.5) v(z) =

(∑n>0

c0,me−mπz,

∑n>0

c1;ne−nπz

),

where c0;m is as in (8.3) and c1;n ∈ R2 all n ∈ Z>0. If v([0,∞) × 0) ⊂ LK andv([0,∞)× 1) ⊂ LK then

(8.6) v(z) =

(∑n>0

c0,me−mπz, i

∑n>0

c1;ne−nπz

),

where notation is as in (8.5). We say that the smallest half-integer n + 1/2 in (8.3)or (8.4) such that c1,n+1/2 6= 0 or the smallest integer n in (8.5) or (8.6) such thatc1;n 6=0 is the asymptotic winding number of u at its Lagrangian intersection puncture.

8.4. Holomorphic disks. — Consider the closed unit diskD ⊂ C withm+1 cyclicallyordered distinct points z0, . . . , zm on ∂D. Set D := D r z0, . . . , zm. Consider aJ-holomorphic map u : D → R × S∗Q resp. T ∗Q which maps ∂D r z0, . . . , zm toR×ΛK resp. L = Q∪LK and which has finite ω-energy and λ-energy. Proposition 8.10shows that near each puncture zj the map u either extends continuously, or it ispositively or negatively asymptotic to a Reeb chord. We will use the following notationfor such disks.

A symplectization disk (with m > 0 negative punctures) is a J-holomorphic map

u : (D, ∂D) −→ (R× S∗Q,R× ΛK)

with positive puncture at z0 and negative punctures at z1, . . . , zm. A cobordism disk(with m > 0 Lagrangian intersection punctures) is a J-holomorphic map

u : (D, ∂D) −→ (T ∗Q,L)

with positive puncture at z0 and Lagrangian intersection punctures at z1, . . . , zm.Let b = b1b2 · · · bm be a word of m Reeb chords. We write

M sy(a, n0; b1, . . . , bm) = M sy(a, n0; b)

for the moduli space of symplectization disks with positive puncture asymptotic tothe Reeb chord a where the order of convergence is n0 and m negative punctures(in counterclockwise order) asymptotic to the Reeb chords b1, . . . , bm. Here the pointsz0, . . . , zm on ∂D are allowed to vary and we divide by the action of Möbius transfor-mations on D. Note that R acts by translation on these moduli spaces.

J.É.P. — M., 2017, tome 4


Similarly, let n = (n1, . . . , nm) be a vector of half-integers or integers. We write

M (a, n0;n1, . . . , nm) = M (a, n0;n)

for the moduli space of cobordism disks with positive puncture asymptotic to the Reebchord a with degree of convergence n0 and m > 0 Lagrangian intersection punctureswith asymptotic winding numbers given by the integers or half-integers nj . Note thatthe number of half-integers must be even for topological reasons (at each half-integerthe boundary of u switches from Q to LK or vice versa).

In both cases when n0 = 0 we will suppress it from notation and simply write

M sy(a; b) and M (a;n),

respectively.For a Reeb chord c : [0, T ]→ S∗Q of length T , the map uc : R× [0, 1]→ R× S∗Q

given by uc(s + it) = (Ts, c(Tt)) is a J-holomorphic parametrization of R × c andthus a symplectization disk with positive and negative puncture asymptotic to c. Wecall it the Reeb chord strip over c.

8.5. Compactness in R × S∗Q and T ∗Q. — In this subsection we review the com-pactness results proved in [5] that concern compactness of the moduli spaces of holo-morphic disks discussed in Section 8.4.

Let us denote by a source disk Dm the unit disk with some number m + 1 > 1

of punctures z0, . . . , zm on its boundary; we call z0 the positive and z1, . . . , zm thenegative punctures. A broken source disk Dm with r > 1 levels with m+ 1 boundarypunctures is represented as a finite disjoint union of punctured disks,

Dm = D1,1 ∪ (D2,1 ∪ · · · ∪D2,`2) ∪ · · · ∪ (Dr,1 ∪ · · · ∪Dr,`r ),

where (Dj,1 ∪ · · · ∪Dj,`j ) are the disks in the jth level and we require the followingproperties:

– Each negative puncture q of a disk Dj,k in the jth level for j < r is formallyjoined to the positive puncture of a unique disk Dj+1,s in the (j + 1)th level. We saythat Dj+1,s is attached to Dj,k at the negative puncture q.

– The total number of negative punctures on level r is m.Note that a broken source disk with one level is just a source disk.

We consider first compactness for curves in the symplectization. Let Dm be a bro-ken source disk as above. A broken symplectization disk with r levels with domain Dm

is a collection v of J-holomorphic maps vj,k defined on Dj,k with the following prop-erties:

– For each 1 6 j 6 r and 1 6 k 6 `j , vj,k represents an element in

M sy(aj,k; bj,k1 , . . . , bj,ks ).

Moreover, for j > 1, the Reeb chord aj,k at the positive puncture of vj,k matches theReeb chord bj−1,k′ at the negative puncture of vj−1,k′ in Dj−1,k′ at which Dj,k isattached.

– For each level 1 6 j 6 r, at least one of the maps vj,k is not a Reeb chord strip.

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An arc in a source disk is an embedded curve that intersects the boundary only atits end points and away from the punctures. We say that a sequence of symplectizationdisks

uj ⊂M sy(a; b1, . . . , bm)

converges to a broken symplectization disk if there are disjoint arcs γ1, . . . , γk in thedomains of uj which give the decomposition of the domain into a broken source diskin the limit and such that in the complement of these arcs, the maps uj converge tothe corresponding map of the broken disk uniformly on compact subsets.

Theorem 8.11. — Any sequence uj ⊂ M sy(a, b1, . . . , bm) of symplectization diskshas a subsequence which converges to a broken symplectization disk v with r > 1 levels.

Proof. — Follows from [3] (see also [5, Th. 1.1]).

In order to describe the compactness result for moduli spaces of holomorphic disksin T ∗Q we first introduce a class of constant holomorphic disks and then the notionof convergence to a constant disk. A constant holomorphic disk is a source disk Dm,m > 3, a constant map into a point q ∈ K, and the following extra structure: Eachboundary component is labeled by LK or by Q and at each puncture zj there isan asymptotic winding number nj ∈ 1

2 , 1,32 , . . . such that nj is a half-integer if

the adjacent boundary components of Dm are labeled by different components ofL = LK ∪Q and an integer otherwise, and such that n0 =

∑mj=1 nj .

A sequence of holomorphic maps vj : Dm → T ∗Q with boundary on L convergesto a constant holomorphic disk if it converges uniformly to the constant map on anycompact subset and if for all sufficiently large j, vj takes any boundary componentlabeled by LK or Q to LK or Q, respectively, and if the asymptotic winding numbersat the negative punctures of the maps vj agree with those of the constant limit mapat corresponding punctures.

Let Dm be a broken source disk with r levels and suppose 1 6 r0 6 r. A bro-ken cobordism disk with r0 non-constant levels and domain Dm is a collection v ofJ-holomorphic maps vj,k defined on Dj,k with the following properties.

– For j < r0 and 1 6 k 6 `j , vj,k represents an element in

M sy(aj,k; bj,k1 , . . . , bj,ks ).

Moreover, for j > 1, the Reeb chord aj,k at the positive puncture of vj,k matches theReeb chord bj−1,k′ at the negative puncture of vj−1,k′ in Dj−1,k′ at which Dj,k isattached.

– For each level j < r0, at least one of the maps vj,k is not a Reeb chord strip.– For j = r0 and 1 6 k 6 `j , vj,k represents an element in

M (aj,k;nj,k1 , . . . , nj,ks )

and the Reeb chord at the positive puncture of vj,k matches the Reeb chord at thenegative puncture of vj−1,k′ in Dj−1,k′ at which Dj,k is attached.

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– For j > r0, vj,k is a constant map to q ∈ K, where q ∈ K is the image of thenegative puncture of vj−1,k′ in Dj−1,k′ , at which Dj,k is attached. Moreover, Dj,k

has at least 3 punctures and the winding number and labels at its positive punctureagree with those of the negative puncture where it is attached. (From the point ofview of the source disk these constant levels encode degenerations of the conformalstructure corresponding to colliding Lagrangian intersection punctures, see Section10.3 for more details.)We say that the disks in levels j < r0 are the symplectization disks, that the disks inlevel r0 are the cobordism disks, and that disks in levels j > r0 are the constant disksof the broken disk.

We define convergence to a broken cobordism disk completely parallel to the sym-plectization case.

Theorem 8.12. — Let uj ⊂ M (a;n1, . . . , nm) be a sequence of cobordism disks.Then uj has a subsequence which converges to a broken cobordism disk.

Proof. — This is a consequence of [5, Th. 1.1]. Note that the levels of constant disksare recovered by the sequence of source disks that converges to a broken source disk.

Remark 8.13. — We consider the convergence implied by the Compactness Theorem8.12 in more detail in a special case relevant to the description of our moduli spacesbelow. Consider a sequence of holomorphic disks uj as in the theorem that convergesto a broken cobordism disk with top level v and such that all disks on lower levels areconstant. Let q` be a negative puncture of the top level v and let D` be the (possiblybroken) constant disk attached with its positive puncture at q`.

Consider the sequence of domains of uj as a sequence of strips with slits Sj , see thediscussion of standard domains in Section 9.1 and Figure 9.1. It follows from the proofof [5, Th. 1.1] that there is a strip region [−ρj , 0]×[0, 1] ⊂ Sj , where ρj →∞ as j →∞such that in the limit the negative puncture q` of v corresponds to (−∞, 0] × [0, 1]

and the positive puncture of the domain D` corresponds to [0,∞) × [0, 1] attachedat this puncture. Assume that q` maps to x ∈ K and consider the Fourier expansionof v near q` in the local coordinates near K perpendicular to the knot:

v(s+ it) = ek0π(s+it)∞∑k=0

ckekπ(s+it),

where k0 > 1/2 is a half-integer and ck are vectors in R2 or iR2, c0 6= 0. We say thatthe complex line spanned by c0 is the limiting tangent plane of v at q`. Writing vusing Taylor expansion as a map from the upper half plane with the puncture q` atthe origin and taking the complex line of c0 as the first coordinate we find that thenormal component of v at x is given by

v(z) =(zk0 ,O(zk0+1)

),

after suitable rescaling of the first coordinate.

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We next restrict to the case relevant to our applications, of a sequence of disks ujwith a constant disk with three or four punctures splitting off. The three punctureddisk is simpler, so we consider the case of a disk with four punctures splitting off. Inthis case, consider a vertical segment ρ0×[0, 1] in the stretching strip [−ρj , 0]× [0, 1].It subdivides the domain of uj in two components D+ containing the positive punc-ture and its complement D−. Consider the Fourier expansion of uj near this verticalsegment. We have

uj(s+ it) =∑k>k0

cj;ke−kπ(s+it),

where k are half-integers and cj;k ∈ R2 (or iR2). Since the winding number alongthe vertical segment is equal to the sum of the winding numbers of the negativepunctures in the component of D− that it bounds, we find that, for j sufficientlylarge, cj;k = 0 for all k < 3/2, hence k0 > 3/2. Moreover, cj;k0 converges to a vectorin the limiting tangent plane of u0 at the newborn negative puncture. In the genericcase, see Lemma 9.5, this limiting vector is non-zero. We assume for definiteness inwhat follows that it is equal to (1, 0).

Pick a conformal map taking D− to the half disk of radius 1 in the upper halfplane, with the vertical segment corresponding to the half circular arc and with themiddle boundary puncture mapping to 0. Then as j → ∞ the locations of the othertwo punctures both converge to 0 and, for large j, the projection to the first complexcoordinate determines the location of the other two punctures. Moreover, the sumof the winding numbers at these three punctures equals 3

2 (i.e., the winding numberalong the half circle of radius 1). Consequently, we have, with z a coordinate on theupper half plane, for all j large enough

uj(z) =√z(z − δj)(z − εj)

((1, 0) + vj + O(z)

),

where δj , εj → 0 and vj → 0 ∈ R2 as j → ∞. It follows that disks in a limitingsequence eventually lie close to the model disk (4.2) discussed in Section 4.3.

There is a completely analogous and simpler analysis of the case when two punc-tures collide which shows that disks in a limiting sequence are close to the model disk(4.1) of Section 4.3 in the same sense.

9. Transversely cut out solutions and orientations

In this section we show that the moduli spaces in Section 8 are manifolds forgeneric almost complex structure J . To accomplish this, we first express each modulispace as the zero locus of a section of a bundle over a Banach manifold and thenshow, using an argument from [15], that one may make any section transverse tothe 0-section by perturbing the almost complex structure. Here cases of disks withunstable domains require extra care: we stabilize their domains using extra markedpoints on the boundary. We control these marked points using disks with higher orderof convergence to Reeb chords.

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Figure 9.1. A standard domain.

9.1. Conformal representatives and Banach manifolds. — In order to define suit-able Banach spaces for our study of holomorphic curves we endow the domains of ourholomorphic disks with cylindrical ends. For convenience we choose a particular suchmodel for each conformal structure on the punctured disks. (The precise choice is notimportant since the space of possible choices of cylindrical ends is contractible.)

– A standard domain ∆0 with one puncture is the unit disk in the complex planewith a puncture at 1 and fixed cylindrical end [0,∞)× [0, 1] at this puncture.

– A standard domain ∆1 with two punctures is the strip R× [0, 1].– A standard domain ∆m([a1, . . . , am−1]) with m + 1 > 2 boundary punctures is

a strip R × [0,m] ⊂ C with slits of small fixed width (and fixed shape) around half-infinite lines (−∞, aj ]×j, where 0 < j < m is an integer, removed. See Figure 9.1.We say that aj ∈ R is the jth boundary maximum of ∆m([a1, . . . , am−1]).

The space of conformal structures Cm on the (m+1)-punctured disk is then repre-sented as Rm/R where R acts on vectors of boundary maxima by overall translation,see [9, §2.1.1]. The boundary of the space of conformal structures on an (m + 1)-punctured disk in its compactification ∂Cm ⊂ Cm can then be understood as consist-ing of the several level disks which arise as some differences |aj−ak| between boundarymaxima approach∞. We sometimes write ∆m for a standard domain, suppressing itsconformal structure [a1, . . . , am−1] from the notation.

The breaking of a standard domain into a standard domain of several levels iscompatible with the compactness results Theorems 8.11 and 8.12. In the proof ofthese results given in [5], after adding a finite number of additional punctures thederivatives of the maps are uniformly bounded and each component in the limithas at least two punctures and can thus be represented as a standard domain. Inparticular, the domain right before the limit is the standard domain obtained bygluing these in the natural way and the arcs in the definition of convergence canbe represented by vertical segments. Here a vertical segment in a standard domain∆m ⊂ C is a line segment in ∆m parallel to the imaginary axis which connects twoboundary components of ∆m.

9.2. Configuration spaces. — In this section we construct Banach manifolds whichare configuration spaces for holomorphic disks. In order to show that all moduli spaceswe use are manifolds we need to stabilize disks with one and two punctures by addingpunctures in a systematic way. To this end we will use Sobolev spaces with extra

J.É.P. — M., 2017, tome 4


weights. This is the reason for introducing somewhat more complicated spaces below.The constructions in this section parallels corresponding constructions in [15] and [19].

We first define the configuration space for holomorphic disks in T ∗Q and then findlocal coordinates for this space showing that it is a Banach manifold. We then repeatthis construction for disks in the symplectization.

Below we are interested in the moduli spaces M (a, n0;n) of holomorphic disks forn0 = 0 or n0 = 1 and n = (n1, . . . , nm), which we will describe as subsets of suitableconfiguration spaces W = W (a; δ0;n). Here δ0 > 0 and n0 are related as follows:Consider the standard neighborhood (−ε, T + ε)×U (with U ⊂ C2) of the Reeb corda : [0, T ] → S∗Q which we introduced on page 739. The projections of the contactplanes at the two end points of a to C2 intersect transversally, and we denote by0 < θ′π 6 θ′′π < π the two complex angles between them. Now for n0 = 0 we choose0 < δ0 < θ′ and for n0 = 1 we choose θ′′ < δ0 < 1.

The space W fibers over the product space

B = Rm−2 × R× J(K).

The first factor Rm−2 is the space of conformal structures on the disk with m + 1

boundary punctures. We represent the disk as a standard domain with the first bound-ary maximum at 0 and Rm−2 as the coordinates of the remaining m − 2 boundarymaxima. The second factor R corresponds to the shift in parameterization of theasymptotic trivial strips at the positive puncture. The third factor is itself a productwith one factor for each negative puncture:

J(K) = J (r1)(K)× · · · × J (rm)(K).

Here rj is the smallest integer < nj and J (rj)(K) denotes the rthj jet-space of K.

A point (q0, q1, . . . , qrj ) ∈ J (rj)(K) corresponds to the first Fourier (Taylor) coef-ficients of the map at the jth negative puncture. Note that J(K) depends on n =

(n1, . . . , nm), but we omit this dependence from the notation.Fix a parameterization of each Reeb chord strip. If γ ∈ Rm−2 then we write

∆[γ] for the standard domain with first boundary maximum at 0 and the followingboundary maxima according to the components of γ. If n = (n1, . . . , nm) ∈ ( 1

2Z)m

with∑j nj ∈ Z then we decorate the boundary components of ∆[γ] according to n

as follows. Start at the positive puncture and follow the boundary of ∆[γ] in thepositive direction. Decorate the first boundary component by LK and then when wepass the jth negative puncture we change Lagrangian (from LK to Q or vice versa)if nj is a half integer and do not change if it is an integer.

Fix a smooth family of smooth maps

wβ : (∆[β1], ∂∆[β1]) −→ (T ∗Q,L), β = (β1, β2, β3) ∈ B,

with the following properties:– wβ respects the boundary decoration, i.e., it takes boundary components deco-

rated by LK resp. Q to the corresponding Lagrangian submanifold.

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– wβ agrees with the Reeb chord strip of a shifted by β2 in a neighborhood of thepositive puncture.

– Consider standard coordinates C×C2 near the first component of βj3 ∈ J (rj)(K).Then in a strip neighborhood of the jth negative puncture, the C2-component of wβvanishes and the C-component is given by

wβ(z) =

rj∑l=0

qlelπz,

where the jth component βj3 of β3 is

βj3 = (q0, q1, . . . , qrj ) ∈ J (rj)(K).

Let 0 < δ < 1/2 and as before let either 0 < δ0 < θ′ or θ′′ < δ0 < 1, where θ′describes the smallest non-zero complex angle at the Reeb chord a and θ′′ the largest.Let Hδ0,δ(β1) denote the Sobolev space of maps

w : ∆[β1] −→ T ∗R3 ∼= R6

with two derivatives in L2 and finite weighted 2-norm with respect to the weightfunction ηδ with the following properties.

– ηδ0,δ equals 1 outside a neighborhood of the punctures.– ηδ0,δ(s+ it) = eδ0π|s| near the positive puncture.– ηδ0,δ(s+ it) = e(nj−δ)π|s| near the jth negative puncture.Consider the bundle E → B with fiber over β ∈ B given by Hδ0,δ(β1). Define the

configuration space W = W (a; δ0;n) ⊂ E of (β,w) such that u = wβ +w satisfies thefollowing

– u takes the boundary of ∆[β1] to L respecting the boundary decoration.– u is holomorphic on the boundary, i.e., the restriction (trace) of ∂Ju to ∂∆[β1]

vanishes.It is not hard to see that W is a closed subspace of E. In fact it is a Banach sub-

manifold of the Banach manifold E. We will next explain how to find local coordinateson W . Let (β,w) ∈ E, and assume that u = wβ + w is a map in W .

In order to find local coordinates around u we first consider the finite dimensionaldirections. Pick diffeomorphisms of the source ∆[β1],

(9.1) φγ , γ ∈ R; ψη1 , η1 ∈ Rm−1,

corresponding to the second and first finite dimensional factors. Here φγ equalsthe identity outside a neighborhood of the positive end where it equals translationby γ, and ψη1 : ∆[β1] → ∆[β1 + η1] moves the boundary maxima according to η1,see [9, §6.2.3].

We next turn to the translations along the knot and the infinite dimensional com-ponent of the space. Using the coordinate map of Lemma 8.6 we import the flatmetric on T ∗(S1 ×D2) to T ∗Q, we extend this metric to a metric h1 on all of T ∗Qso that LK is totally geodesic and flat near Reeb chord endpoints, see 8.1 (v), andsuch that h1 = ds2 + g on T ∗Q r D∗Q ∼= R+ × S∗Q, where g is a metric on S∗Q.

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Consider the standard almost complex structure in a neighborhood of the zero sec-tion of Q = R3 in T ∗Q. Note that this almost complex structure agrees with thestandard almost complex structure in the holomorphic neighborhood of K. Using theconstruction in [13, Prop. 5.3], we extend it to an almost complex structure J over allof T ∗Q with the following additional property near LK . If V is a vector field along ageodesic in the metric h1 in LK then V satisfies the Jacobi equation if and only if thevector field JV does. To achieve this we might have to be alter h1 slightly near butnot on LK , see [13, Eq. (5.7)] for the precise form of h1 (corresponding to g in thatequation). Note that this construction gives the standard almost complex structurenear the knot. Let h0 denote the standard flat metric on T ∗Q and note that it hasthe Jacobi field property discussed above along Q. Let

(9.2) hσ, 0 6 σ 6 1

be the linear interpolation between the metrics h0 and h1.Consider the pullback bundle u∗T (T ∗Q). Note that the Riemannian metrics ht on

T ∗Q induce connections on this bundle which we denote by ∇t.Let ˙Hδ(u) denote the linear space of sections v of u∗T (T ∗Q) with the following

properties.– The partial derivatives of v up to second order lie in L2

loc(∆[β], u∗T (T ∗Q)).– The restriction of ∇σv+J ∇σv i to the boundary (sometimes called the trace

of ∇σv+J ∇σvi) vanishes, where σ = 1 for a boundary component mapping to LKand σ = 0 for a component mapping to Q.

– With ‖ · ‖δ,δ0,n denoting the Sobolev 2-norm weighted by ηδ,δ0,n, ‖v‖δ,n <∞.Then ˙H2,δ,δ0,n(w) equipped with the norm ‖ · ‖δ,δ0,n is a Banach space.

Also fix m +∑mj=1 rj smooth vector fields sjk, 1 6 j 6 m and 0 6 k 6 rj along u

with properties as above and with the following additional properties.– The vector field sjk is supported only near the jth negative puncture in a half

strip neighborhood which maps into the analytic neighborhood of the knot.– In standard coordinates along the knot C×C2, the C2-component of sjk equals 0

and the C-component is sjk = ekπz.We are now ready to define the local coordinate system. Write expσ for the expo-

nential map in the Riemannian metric hσ, 0 6 σ 6 1, from (9.2). The local coordinatesystem around u has the form

Φu : U1 × U2 × U3 ×U −→ W ,

where U1 ⊂ Rm−2, U2 ⊂ R, U3 ⊂ Πmj=1Rrj+1, and U ⊂ ˙H2,δ,δ0,n(w) are small

neighborhoods of the origin with coordinates γj ∈ Uj . Let σ : ∆[β1] → [0, 1] be asmooth function that equals 0 resp. 1 in a neighborhood of any boundary componentthat maps to Q resp. LK and that equals 0 on u−1(D∗Q). For u as above we thenconsider

Ψu(γ1, γ2, γ3, v)(z) = expσ(z′)u(z′)

(v(z′) +

m∑j=1

rj∑k=0

γ3jksjk(z′)

), z′ = φγ1(ψγ2(z)),

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see (9.1) for the diffeomorphisms φγ1 and ψγ2 . Here γ1 corresponds to shifts near thepositive puncture, γ2 corresponds to variations of the conformal structure, γ3 is relatedto variations of the map near Lagrangian intersection punctures, and v is a vectorfield along the curve. We use the exponential map to go from linearized variations toactual maps.

Lemma 9.1. — The space W is a Banach manifold with local coordinates around ugiven by Ψu.

Proof. — This is straightforward, see [15, Lem. 3.2] for an analogous result.

Consider the bundle E over the configuration space W with fiber over u the complexanti-linear maps

T∆[β1] −→ T (T ∗Q).

The ∂J -operator gives a section of this bundle u 7→ (du+ J du i) and the modulispace M (a;n0,n) is the zero locus of this section, where n0 = 0 if 0 < δ0 < θ′ andn0 = 1 if θ′′ < δ0 < 1. The section is Fredholm and the formal dimension of thesolution spaces is given by its index. We have the following dimension formula.

Lemma 9.2. — The formal dimension of M (a, n0;n) is given by

dim(M (a, n0;n)) = |a| − 2n0 − |n|.

Proof. — The case n0 = 0 follows from [5, Th.A.1 & Rem.A.2]. The fact that theindex jumps when the exponential weight crosses the eigenvalues of the asymptoticoperator is well known and immediately gives the other case, see e.g. [13, Prop. 6.5].

We next consider a completely analogous construction of a configuration space forholomorphic disks in M sy(a, n0; b). We discuss mainly the points where this con-struction differs from that above. Consider first the finite dimensional base. Here thesituation is simpler and we take instead

B = Rm−2 × Rm+1,

where the first factor corresponds to conformal structures on the domain exactly asbefore and where the second factor corresponds to re-parameterizations of the trivialReeb chord strips exactly as for the positive puncture before. We fix a smooth familyof maps wβ : ∆[β1] → R × S∗Q which agrees with the prescribed Reeb chord stripsnear the punctures. We next fix an isometric embedding of S∗Q into RN and considerthe bundle of weighted Sobolev spaces with fiber over β ∈ B the Sobolev space Hn0,δ

of functions with two derivatives in L2 with respect to the norm weighed by a functionwhich equals eδ|s| in the negative ends and e(δ+n0)|s| in the positive end.

In analogy with the above we then fix (commuting) re-parameterization diffeomor-phisms ψβ1 corresponding to changes of the conformal structure and φβ2 correspond-ing to translation in the half strip neighborhoods. Again this then leads to a Fredholmsection and its index gives the formal dimension of the moduli space.

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Lemma 9.3. — The formal dimension of M sy(a, n0; b) is given by

dim M sy(a, n0; b) = |a| − 2n0 − |b|.

Proof. — See [5, Th.A.1 & Rem.A.2] and use the relation between weights and index,see e.g. [13, Prop. 6.5].

Remark 9.4. — We consider for future reference the conformal variations of the do-main with more details. In the local coordinates around a map w : ∆m+1 → T ∗Q

or w : ∆m+1 → R × S∗Q defined above, the conformal variations correspond to adiffeomorphism that moves the boundary maxima of the domain. We take such adiffeomorphism to be a shift along a constant (and hence holomorphic) vector field τin the real direction around the boundary maximum and then cut it off in nearbystrip regions. Hence the corresponding linearized variation L∂J(γ) at w, where γ isthe first order variation of the complex structure corresponding in the domain is

L∂J(γ) = ∂Jw ∂τ.

We will sometimes use other ways of expressing conformal variations, where thevariations are supported near a specific negative puncture rather than near a specificboundary maximum. To this end we first note that we may shift the conformal varia-tion by any element L∂J(v) where v is a vector field along w in the Sobolev space Hδ.In particular we can shift γ by ∂σ where σ is a vector field along ∆m+1 that is con-stant near the punctures. In this way we get equivalent conformal variations γq of theform

L∂J(γq) = ∂Jw ∂τq.where τq is a vector field of the form

τq(z) = β(s+ it)eπ(s+it),

where s+ it is a standard coordinate in the strip neighborhood of the negative punc-ture a and β is a cut-off function equal to 1 near the puncture and 0 outside a stripneighborhood of the puncture. We refer to [9, §2.1.1] for details.

9.3. Transversality. — We next use the special form of our almost complex structurenear Reeb chords in combination with an argument from [15, Lem. 4.5] to show thatwe can achieve transversality for ∂J -section of E over W by perturbing the almostcomplex structure. In other words we need to show that the linearization L∂J of thesection ∂J is surjective.

Lemma 9.5. — For generic J any solutions in M (a, n0;n) and M sy(a, n0; b) aretransversely cut out.

Proof. — To see this we perturb the almost complex structure near the positive punc-ture. Consider the local projection to C2 near the Reeb chord. Here the Lagrangianscorrespond to two Lagrangian planes. Furthermore the holomorphic disks admit localTaylor expansions near the points that map to their intersection. The lemma nowfollows from the proof of [15, Lem. 4.5]. We sketch the argument.

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Let U denote a neighborhood of the Reeb chord strip Ca of a for M sy or of theReeb chord strip in T ∗Q−D∗Q for M . If u is a holomorphic disk then u−1(U ∩Ca)

is the pre-image under u composed with the projection to C2 of the intersection pointof the two Lagrangian planes. It follows by monotonicity that the preimage is a finitecollection of points q0, q1, . . . , qr, where q0 is the positive puncture. If qi is an interiorpoint, let Ei denote a small disk around qi, if qj is a boundary point let Ej denote ahalf-disk neighborhood of qi. If the map u has an injective point near the double pointthen a standard argument perturbing the almost complex structure there establishesthe necessary transversality. We therefore assume that this is not the case. Considerthe image of a small half disk E0 near the positive puncture q0, and note that theboundary arcs end at the positive punctures. Since the map is not injective thereare neighborhoods (after renumbering) E1, . . . , Em where u agrees with the image γunder u of one of the boundary arcs of E0. By analytic continuation, the images ofthese neighborhoods then intersect the Lagrangian sheet of the boundary arc γ′ thatcontains γ. Consequently, the map has multiplicity m+ 1 along γ and multiplicity malong γ′ − γ. Consider a vector field in the cokernel of the linearized operator L∂.Perturbing the almost complex structure near γ′ − γ we see that the contributionsfrom the anti-holomorphic cokernel vector field on E1, . . . , Em must vanish. By uniquecontinuation, the contributions from E1, . . . , Em must then also cancel along γ and itfollows that there is nothing that cancels the perturbation in E0 (just as if the mapwas injective in E0). The desired transversality follows.

9.4. Stabilization of domains. — For disks with more than three punctures thetransversality results in Section 9.3 directly give the solution spaces the structureof C1-smooth manifolds. For the case of unstable domains this is not as direct sincethe solutions admit re-parameterizations that do not act with any uniformity on theassociated configuration spaces. This is a well-known phenomenon and we resolve theproblem by a gauge fixing procedure, adding marked points near the positive punc-ture. This construction was studied in detail in [18, App.A.2] and in [19, §5.2 and 6]and we will refer to these articles for details.

As we shall see below we need only consider moduli spaces of dimension 6 2.Recall the neighborhood a × U , U ⊂ C2 of the Reeb chord a, see the discussionbefore Definition 8.1 on page 739, and the corresponding Fourier expansion of theC2-component of any holomorphic disk near a, see (8.2) on page 745.

Consider a space M (a,n) of formal dimension 6 1. Then by Lemmas 9.2and 9.5 the corresponding space M (a, 1;n) is empty. Consequently, for any solutionu ∈ M (a,n), the first Fourier coefficient of the C2-component of the map near a isnon-vanishing. Let S0;ε and S1;ε be spheres in ΛK of radii ε > 0 around the Reebchord endpoints of a. Non-vanishing of the first Fourier coefficient in combinationwith compactness then implies that for each solution u there are two unique pointsin the boundary of the domain closest to the positive puncture that map to Sj;ε,j = 0, 1, see Figure 9.2. We add punctures at these points. More precisely, weconsider standard domains with two more punctures and require that the maps are

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asymptotic to points in Sj;ε at the extra punctures. In the above notation thesewould be “Lagrangian intersection punctures” in Sj;ε of local winding number 1 inthe direction normal to Sj;ε. The transversality result 9.5 holds as before also for thesolution spaces with extra punctures, so that they are C1-manifolds. The asymptoticproperties above then imply that the solutions with extra punctures capture allholomorphic disks.

Consider next a space M sy(a; b) of formal dimension 6 2. Since any holomorphiccurve in the symplectization can be translated we find that the corresponding spaceM sy(a, 1; b) is again empty and we get a manifold structure by adding two markedpoints near the Reeb chord endpoints exactly as above.

It remains then to consider the case of spaces M (a;n) of formal dimension 2. Herethe corresponding space M (a, 1;n) has dimension 0. There are then a finite num-ber of solutions with this decay condition. Considering the Fourier expansion we canfix unique marked points for all solutions in a neighborhood V (in the configurationspace) of these isolated solutions as above. For solutions outside V the Fourier coeffi-cients do not vanish and we can fix marked points as above. Note however, that thesewill generally not be the same marked points. This way we however get two types ofmanifold charts: one for solutions inside V and one for solutions in a neighborhoodof any map u′ with nonvanishing first Fourier coefficient which lies outside a smallerneighborhood V ′ of u. To get a manifold structure for the moduli space we thenneed to study the transition maps, and to that end we use four marked points, seeFigure 9.2 and [19, §5.2] for details.

Figure 9.2. Top left: marked points for a disk near the disk withdegenerate asymptotics. Top right: marked points for a disk out-side a neighborhood of the disk with degenerate asymptotics. Lower:the four marked points in the intermediate region used to definethe coordinate change. The (black) lines represent the projectionsof the branches of LK ∪ Q to C2, and the (blue) circles represent3-spheres S3

ε , which cut these local branches along the circles Sj,εappearing in the text.

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A priori, the smooth structures on the moduli spaces above depend on the choiceof gauge condition. However, using the fact that the C0-norm of a holomorphic mapcontrols all other norms, it is not hard to see that different gauge conditions lead tothe same smooth structure.

We also need to show that the compactness result where sequences of curves con-verge to several level curves are compatible with additional marked points. This issimilar to the above. The compactness result we already have implies uniform con-vergence on compact sets and in particular it is possible to add marked points onthe curves near the limit that correspond to the extra marked points on the unstablecurves in the limit. As before we show that these extra marked points do not affect themoduli spaces. See [19, §A.3] for details. In conclusion, by adding marked points alsoon curves near broken limits we obtain versions of the compactness results Theorems8.11 and 8.12 where all domains involved are stable with marked points compatiblewith the several level breaking.

9.5. Index bundles and orientations. — Viewing the ∂J -operator as a Fredholmsection of a Banach bundle, its linearization defines an index bundle over the configu-ration space and an orientation of this index bundle gives an orientation of transversesolution spaces. Following Fukaya, Oh, Ohta, Ono [22, §8.1] one defines a coherentsystem of such orientations as follows. Fix spin structures of the two Lagrangians LKand Q, which we here can think of as trivializations of the respective tangent bun-dles. Consider a closed disk with boundary in one of the two Lagrangians and thelinearized ∂J -operator acting on vector fields along this disk that are tangent to theLagrangian along the boundary. Using the trivialization of the boundary condition,such an operator can be deformed to an operator on the disk with values in C3 andconstant R3 boundary condition, with a copy of CP 1 attached at the center witha complex linear operator. The first operator has trivial cokernel and a kernel thatconsists only of constant vector fields, and the orientation of C induces an orientationon the determinant bundle of the operator over CP 1. This gives a canonical orienta-tion over closed disks with trivialized boundary condition (that depends only on thehomotopy class of the trivialization).

Here we need to orient moduli spaces of disks with punctures. This was done inthe setting of Legendrian contact homology in [14]; we will give a sketch and referto that reference for details. We reduce to the case of closed disks by picking so-called capping operators at all Reeb chords and along the Lagrangian intersection Kwith an orientation of the corresponding determinant bundles. Here it is importantthat the capping operators are chosen in a consistent way. At Reeb chords there is apositive and a negative capping operator and we require that they glue to the standardorientation on the closed disk. We also pick positive and negative capping operatorsat the Lagrangian intersection punctures satisfying the same conditions. Now, given aholomorphic disk in the symplectization or in T ∗Q we glue the capping operators to itand produce a closed disk. The standard orientation of the closed disk and the chosenorientations on the capping operators then give an orientation of the determinant line

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of the linearized operator over the disk, which, together with an orientation of thefinite dimensional space of conformal structures on the punctured disk, in turn givesan orientation of the moduli space if it is transversely cut out. The gluing conditionfor the capping operators ensures that the resulting orientations of the moduli spacesare compatible with splittings into multi-level curves.

In what follows we assume that spin structures on the Lagrangians and cappingoperators have been fixed and thus all our moduli spaces are oriented manifolds.

9.6. Signs and the chain map equation. — Recall the chain map

Φ: (C∗(R), ∂Λ) −→ (C∗(Σ), ∂ + δQ + δN )

from Theorem 6.13. Here we consider the signs of the operations δQ and δN in thisformula. These operations are defined on chains of broken strings by taking the ori-ented preimage of K under the evaluation map. In the map Φ, the oriented chain isgiven by a moduli space of holomorphic disks. In order to deal with the evaluationmaps on such spaces we present them as bundles over Q as follows. Consider first theoperation δQ. Fix a point q ∈ Q and an additional puncture on the boundary that werequire maps to q. Concretely, we work on strips with slits and add a small positiveexponential weight at the puncture mapping to q. Then we consider the bundle of suchmaps over Q when we let q vary in Q. The orientation of this space is induced fromcapping operators as described above. When we consider the corresponding boundarycondition on the closed disk we find a vanishing condition for linearized variationsat the marked point corresponding to the positive exponential weight. Thus if σ de-notes the orientation of the index bundle induced as above, then the orientation onthe bundle with marked point mapping to q is given by the orientation of the formaldifference σ TQ. (The formal difference should be interpreted as in K-theory: thedifference ξ η of two bundles ξ and η is represented by a bundle ζ such that thedirect sum ζ ⊕ η is equivalent to ξ.)

We point out that here and throughout this section orientations depend on orderingconventions, whether the point condition goes before or after the index bundle, etc. Incalculations below we put point conditions after the index bundle, and put the fiberof bundles over Q before the base.

The orientation of the bundle corresponding to a point constraint q varying over Qis then given given by σ TQ ⊕ TQ. Finally, the orientation of the chain given bythe preimage of K under the evaluation map is then

(9.3) σ TQ⊕ TQ⊕ TK TQ = σ ⊕ TK TQ.

In order to show that the chain map equation holds we must then show that thereare choices of capping operators and orientations on Q and N so that this orientationagrees with the boundary orientation of the disk viewed as the boundary in the modulispace of disks with two colliding Lagrangian punctures.

Consider the capping operators cQN and cNQ for such a puncture going from Q

to N and vice versa. These capping operators are standard ∂-operators on a once

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punctured disk D1 acting on C3-valued functions in a weighted Sobolev space thatsatisfy a Lagrangian boundary condition.

We first describe the boundary conditions. For cQN the Lagrangian boundary con-dition λ : ∂D1 → Lag3, where Lag3 denotes the Lagrangian Grassmannian of La-grangian subspaces of C3, starts at the tangent space of Q and ends at the tangentspace of N . For cNQ the boundary condition instead starts at the tangent space of Nand ends at the of Q. More specifically, the tangent spaces of Q and N intersectalong TK and are perpendicular in the normal directions of K. We think of thenormal directions to K as C2 and the tangent spaces of Q and N as iR2 and R2, re-spectively. We take both capping operators cQN and cNQ to fix TK, to be a rotationby π/2 in one of the complex lines normal to the knot, and a rotation by 3π/2 in theother.

We next describe the weights at the puncture in D1. We use a half strip neighbor-hood of the puncture and a Sobolev space with small positive exponential weight δ,0 < δ < π/2, in this strip neighborhood.

The index of the ∂-operator with this boundary condition and weight equals 3, seee.g. [13, Prop. 6.5].

Recall from Section 9.5 that an orientation of the moduli space is induced from thecapping operators together with an orientation on the space of conformal structureson the punctured disk. Here we think of variations of the conformal structure asvector fields moving the punctures along the boundary of the disk. We have onesuch vector field for each puncture which give an additional one dimensional orientedvector space associated to each puncture, see [14, §3.4.1] for details. For simplicity wewrite simply cQN and cNQ for the sum of the index bundles of the capping operatordescribed above and one dimensional conformal variations associated to the respectivepunctures. Thus, in the calculations below cQN and cNQ have index 3 + 1 = 4.

We choose the orientations on Q and N so that the linear transformations betweentangent spaces TQ and TN induced by the Lagrangian boundary conditions of cQNand cNQ take the orientation on Q to that on N and vice versa.

The boundary orientation of the two-level disk (second level constant) is the fiberproduct over K of the orientations of its levels. We view the top level disk as havinga small positive exponential weight at the puncture mapping to K and a cut-off localsolution in the direction of K. In analogy with the above, its orientation is thus givenby σ TQ ⊕ TK. The orientation of the constant disk (which has small negativeweights at its positive puncture) is then σ′ ⊕ cQN ⊕ cNQ, where σ′ is the standardorientation on the closed up boundary condition of the constant three punctured disk.The boundary orientation is thus

(9.4) (σ TQ⊕ TK)⊕ (σ′ ⊕ cQN ⊕ cNQ) TK.

Now choose the orientation on cQN and cNQ so that the orientation of the indexone problem on the constant disk with kernel in direction of the knot induced byσ′ ⊕ cQN ⊕ cNQ is opposite to the orientation of TK. Then the orientation in (9.4) is

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σ⊕TKTQ (there is an orientation change when one permutes the odd-dimensionalsummands TK and TQ), in agreement with (9.3).

For the sign of the operation δN we argue exactly as above replacing Q with N

and we must compare the orientations σ ⊕ TK TN and

(σ TN ⊕ TK)⊕ (σ′ ⊕ cNQ ⊕ cQN ) TK.

Compared to the above the main difference is that the summands cNQ and cQN havebeen permuted. However, as explained above, the index of each of these operatorsis 4, so the orientation remains as before and the positive sign for δN is correct forthe chain map.

10. Compactification of moduli spaces and gluing

In this section we show that the moduli spaces M (a;n) and M sy(a; b)/R admitcompactifications as manifolds with boundary with corners. Furthermore, we describethe boundary explicitly in terms of broken holomorphic disks. The smoothness ofindividual strata of the compactified moduli spaces are governed by the TransversalityLemma 9.5. The Compactness Theorems 8.12 and 8.11 describe disk configurationsin the boundary of the compactification. The main purpose of this section is thus toshow how to glue these configurations on the boundary to curves in the smooth partof the moduli space and thereby obtain boundary charts in the sense of manifolds withboundary with corners. Such gluing theorems were proved before in closely relatedsituations and we will discuss details only when they differ from the standard cases.

We first state the structural theorems in Section 10.1 and then turn to the gluingresults and their proofs in the following subsections.

We work throughout this section with an almost complex structure J so thatLemma 9.5 holds. Furthermore we assume that the domains of all holomorphic disksare stable, which can be achieved by adding marked points as explained in Section 9.4.

10.1. Structure of the moduli spaces. — In this subsection we state the results onmoduli spaces of holomorphic disks. As before there are two cases to consider, disksin the symplectization and disks in the cotangent bundle. The structural results allhave the same flavor. Basically we show that a specified moduli space is a manifoldwith boundary with corners of dimension 6 2, and we describe the boundary strataas well as certain submanifolds important for our study. The proofs of the results arethe main goal for the rest of the section.

Recall from Sections 9.2 and 9.3 (with n0 = 0) that for generic J the moduli spacesM (a;n) and M sy(a; b) are manifolds of dimensions

dim M (a;n) = |a| − |n|, dim M sy(a; b) = |a| − |b|.

Here |a| = ind(a) is the degree of the Reeb chord a (which takes only values 0, 1, 2),and to the vector of local winding numbers n = (n1, . . . , nm) (where the nj are

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Du

a

b1 b2 b3

+

−

−

−

Figure 10.1. Disks u : (D, ∂D)→ (R× S∗Q,R× ΛK) in the symplectization.

positive half-integers or integers) we have associated the nonnegative integer

|n| =m∑j=1

2(nj − 12 ) > 0.

If either n or b is empty, the corresponding contribution to the index formula is 0.If a is a Reeb chord of ΛK ⊂ S∗Q, then 0 6 |a| 6 2. Since J1 is R-invariant, 0-dimensional moduli spaces in the symplectization consists only of Reeb chord strips.Thus the only non-empty moduli spaces M sy(a; b) of dimension dsy are the following(write b = b1 . . . bm), see Figure 10.1:

– [2, 0]sy: If |a| = 2 and |b| = 0 (i.e., |bj | = 0 for all j) then dsy = 2.– [2, 1]sy: If |a| = 2 and |b| = 1 (i.e., |bj | = 0 for all j 6= s and |bs| = 1)

then dsy = 1.– [1, 0]sy: If |a| = 1 and |b| = 0 then dsy = 1.Similarly, the only non-empty moduli spaces M (a;n) of dimension d are the fol-

lowing (write n = n1 · · ·nm), see Figures 10.2, 10.3, 10.4, 10.5:– [2, 0]: If |a| = 2 and all nj = 1

2 , then |n| = 0 and d = 2.– [2, 1]: If |a| = 2 and nj = 1

2 for all j 6= s and ns = 1, then |n| = 1 and d = 1.– [2, 3

2 ]: If |a| = 2 and nj = 12 for all j 6= s and ns = 3

2 , then |n| = 2 and d = 0.– [2, 2]: If |a| = 2 and nj = 1

2 for all j 6= s, t, and ns = nt = 1, then |n| = 2 andd = 0.

– [1, 0]: If |a| = 1 and all nj = 12 , then |n| = 0 and d = 1.

– [1, 1]: If |a| = 1 and nj = 12 for all j 6= s and ns = 1, then |n| = 1 and d = 0.

– [0, 0]: If |a| = 0 and nj = 12 all j, then |n| = 0 and d = 0.

It follows from Theorem 8.12 and Lemma 9.5 (see also Section 9.4) thatthe 0-dimensional moduli spaces listed above are transversely cut out compact0-manifolds. The corresponding structure theorems for moduli spaces of dimensionone and two are the following.

Recall that R acts on holomorphic disks in the symplectization R× S∗Q by trans-lation. Dividing out this action, we obtain moduli spaces of dimension zero and onein the symplectization which have the following structure.

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a

Q

LK LK

12

12

a

Q

LK LK LK

12 1 1

2

[2, 0] : |a| = 2, dim = 2

[1, 0] : |a| = 1, dim = 1

[0, 0] : |a| = 0, dim = 0

[2, 1] : |a| = 2, dim = 1

[1, 1] : |a| = 1, dim = 0

Figure 10.2. Curves with |n| = 0. Figure 10.3. Curves with |n| = 1.

a

Q

LK LK LK

12

32

a

Q

LK LK

12 1 1 1

2

[2, 32 ] : |a| = 2, dim = 0 [2, 2] : |a| = 2, dim = 0

Figure 10.4. Curves with |n| = 2. Figure 10.5. Curves with |n| = 2.

Theorem 10.1. — Moduli spaces of holomorphic disks in the symplectization satisfythe following.

(i) If M sy(a; b) is a moduli space of type [2, 1]sy or of type [1, 0]sy, then M sy(a; b)/Ris a compact 0-manifold.

(ii) If M sy(a; b) is a moduli space of type [2, 0]sy, then M sy(a; b)/R admits a natu-ral compactification M sy(a, b) which is a compact 1-manifold with boundary. Bound-ary points of M sy(a; b) correspond to two-level disks v where the level one disk v1

is of type [2, 1]sy, and where exactly one level two disk v2,s is of type [1, 0]sy and allother level two disks v2,j, j 6= s, are trivial Reeb chord strips.

In the cotangent bundle we have moduli spaces of dimension zero, one, or two. Westart with the 0-dimensional case.

J.É.P. — M., 2017, tome 4


Theorem 10.2. — Moduli spaces M (a;n) of holomorphic disks of types [2, 32 ], [2, 2],

[1, 1], or [0, 0] are compact 0-dimensional manifolds.

In the 1-dimensional case we consider two cases separately. We first consider thecase when |n| = 0.

Theorem 10.3. — Moduli spaces M (a;n) of disks of type [1, 0] admit natural compact-ifications M (a;n) which are 1-manifolds with boundary. Boundary points of M (a;n)

correspond to the following.(a) Two-level disks v where the level one disk v1 has type [1, 1] and where the second

level is a three punctured constant disk v2 attached at the Lagrangian intersectionpuncture of v1 where the asymptotic winding number equals 1.

(b) Two-level disks v where the top level disk v1 is a symplectization disk of type[1, 0]sy and where all the second level disks v2,j, 1 6 j 6 k are of type [0, 0].

(c) If there are no entries in n, then all points of the reduced moduli spaceM sy(a;∅)/R containing disks of type [1, 0]sy appear as boundary points.

In the second 1-dimensional case |n| = 1 and we have the following.

Theorem 10.4. — Moduli spaces M (a;n) of disks of type [2, 1] admit natural compact-ifications M (a;n) which are 1-manifolds with boundary. Boundary points of M (a;n)

correspond to the following.(a) Two-level disks v where the level one disk v1 has type [2, 2] and where the

second level is a three punctured constant disk v2 attached at the new-born Lagrangianintersection puncture of v1 with winding number 1.

(b) Two-level disks v where the top level disk v1 is a symplectization disk of type[2, 1]sy and where the second level consists of disks v2,j, 1 6 j 6 k such that forsome s, v2,s has type [1, 1] and v2,j has type [0, 0] for j 6= s.

(c) Two-level disks v where the level one disk is of type [2, 32 ] and where the second

level disk is a constant three punctured disk attached at the Lagrangian intersectionpuncture with winding number 3

2 . (Here the constant disk has winding number 32 at

its positive puncture, and 1 and 12 at its negative punctures.)

Remark 10.5. — In order to parameterize a neighborhood of the boundary pointsin Theorem 10.3(a) and Theorem 10.4(a) one can use the local model (4.1) fromSection 4.3. Here the location ε > 0 of the puncture on the real axis can be usedas local coordinate for the moduli space. Furthermore, the maps in the moduli spacediffer from the map in (4.1) by terms of order O(z2), so they have a spike that vanishesas ε → 0 as shown at the top of Figure 4.1. Similarly, in order to parameterizea neighborhood of the boundary points in Theorem 10.4 (c) one can use the localmodel (4.2) from Section 4.3 with δ = 0. Here the location ε > 0 of the puncture onthe real axis can be used as local coordinate for the moduli space. Furthermore, themaps in the moduli space differ from the map in (4.2) by terms of order O(z5/2), sothey have a spike that vanishes as ε→ 0 as shown at the bottom of Figure 4.1 (withδ = 0). See Remark 10.16 for details.

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In the 2-dimensional case we have the following description of the structure of themoduli space which is naturally more involved.

Theorem 10.6. — Moduli spaces M (a;n) of disks of type [2, 0] admit natural com-pactifications M (a;n) which are 2-manifolds with boundary with corners. The top-dimensional strata of the boundary have codimension 1 in M (a;n) and correspond tothe following.

(a1) Two-level disks v where the top level disk v1 has type [2, 1] and where the sec-ond level is a three punctured constant disk v2 attached at the Lagrangian intersectionpuncture of v1 where the asymptotic winding number equals 1.

(b1) Two-level disks v where the top level disk v1 is a symplectization disk of type[2, 0]sy and where all the second level disks v2,j, 1 6 j 6 k are of type [0, 0].

(c1) Two-level disks v where the top level disk v1 is a symplectization disk of type[2, 1]sy and where the second level consists of disks v2,j, 1 6 j 6 k such that forsome s, v2,s has type [1, 0] and v2,j has type [0, 0] for j 6= s.

The corner points on the boundary (i.e., the codimension two strata) of M (a;n)

correspond to the following.

(a2) Two-level disks v where the top level disk v1 has type [2, 2] and where thesecond level consists of two three punctured constant disks v2,1 and v2,2 attached atthe Lagrangian intersection punctures of v1 where the winding numbers are 1.

(b2) Three-level disks v where the top level disk v1 is a symplectization disk oftype [2, 1]sy, where the second level disk v2,s is of type [1, 1] and all other second leveldisks v2,j, j 6= s are of type [0, 0], and where the third level consists of a constantthree punctured disk v3 attached at the Lagrangian intersection puncture of v2,s withwinding number 1.

(c2) Three-level disks v where the top level disk v1 is a symplectization disk of type[2, 1]sy, where the second level disk v2,s is of type [1, 0]sy and all other second leveldisks v2,j are Reeb chord strips, and where the third level consists of disks v3,j all oftype [0, 0].

(d2) Two-level disks v where the top level disk v1 has type [2, 32 ] and where the

second level consists of a 4-punctured constant disk v2 attached at the Lagrangianintersection puncture of v1 where the asymptotic winding number is 3

2 .

Remark 10.7. — In order to parameterize a neighborhood of the corner points inTheorem 10.6 (d2) one can use the local model (4.2) from Section 4.3 Here the loca-tions (ε, δ) of the punctures on the real axis can be used as local coordinates for themoduli space. Furthermore, the maps in the moduli space differ from the map in (4.2)by terms of order O(z5/2), so they have two spikes that vanish as ε, δ → 0 as shownat the bottom of Figure 4.1. See Remark 10.17 for details.

Some of the moduli spaces above admit natural maps into others by forgetting someLagrangian intersection punctures. We next describe such maps. It is convenient to

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write12

s= 1

2 ,s. . ., 1

2 .

We consider first the case when the target is a one dimensional moduli space.

Theorem 10.8. — Consider a moduli space M (a; 12

s, 1, 1

2

t) of disks of type [1, 1]. For-

getting the (s+ 1)th Lagrangian intersection puncture we get a map

M (a; 12

s, 1, 1

2

t) −→M (a; 1

2

s+t)

into the compactified moduli space of disks of type [1, 0]. This map is an embedding ofa 0-dimensional manifold into the interior of a 1-manifold.

Finally, we consider similar maps when the target space is two dimensional.

Theorem 10.9. — Consider a compactified moduli space M (a; 12

s, 1, 1

2

t) of disks of

type [2, 1]. Forgetting the (s+ 1)th Lagrangian intersection puncture we get a map

ι : M (a; 12

s, 1, 1

2

t) −→M (a; 1

2

s+t) = M

into the compactified moduli space of disks of type [2, 0]. This map is an immersionof a 1-dimensional manifold into a 2-manifold with boundary with corners. Let M s+1

denote the image of this immersion. Then M s+1 consists of those disks for whichsome point in the (s+1)th boundary arc hits K. Then M s+1 and M t+1 intersect (self-intersect if s = t) transversely at disks with two points hitting K (this correspondsto disks of type [2, 2]). The boundary of M s consists of points in the codimensionone boundary of M corresponding to disks as in Theorem 10.4 (a) and (b) as wellas to interior points corresponding to disks of type [2, 3

2 ] as in Theorem 10.4 (c).Furthermore M s+1 and M s+2 with a common boundary point corresponding to adisk of type [2, 3

2 ] fit together smoothly at this point.

10.2. Floer’s Picard lemma. — In the following subsections we show that the brokendisks in Theorems 10.3–10.4 can be glued in a unique way to give disks in the interiorof the moduli space thus providing a standard neighborhood of the boundary of themoduli space inside the compactified moduli space. Our approach here is standardand starts from Floer’s Picard lemma, see [26] for a proof.

Lemma 10.10. — Let f : B1 → B2 be a smooth map of Banach spaces which satisfies

f(v) = f(0) + df(0)v +N(v),

where df(0) is Fredholm and has a right inverse Q satisfying

(10.1) ‖QN(u)−QN(v)‖ 6 G(‖u‖+ ‖v‖)‖u− v‖

for some constant G. Let B(0, r) be the r-ball centered at 0 ∈ B1 and assume that

(10.2) ‖Qf(0)‖ 6 1

8G.

Then for r < 1/4G, the zero-set f−1(0)∩B(0, r) is a smooth submanifold of dimensiondim(ker(df(0))) diffeomorphic to the r-ball in ker(df(0)).

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We will apply this result as well as a parameterized version of it, see [18, Lem. 5.13].In our case f will be the ∂J -operator. To show existence of solutions near a brokensolution we must thus establish three things: a sufficiently good approximate solu-tion w near the broken solution corresponding to 0 in Lemma 10.10, a right inversefor the linearization of the ∂J -operator at w, corresponding to Q in Lemma 10.10, anda quadratic estimate for the non-linear term in the Taylor expansion, correspondingto (10.1). Here the Banach space B1 will be a product of a weighted Sobolev spaceand a certain finite dimensional space that will serve as a neighborhood of the brokenconfiguration and the Banach space B2 will be a space of fields of complex antilinearmaps. In addition to verifying uniform invertibility of the differential and the non-linear estimate we must also check that the gluing construction captures all solutionsnear the broken solution and that the natural change of coordinates (from the Banachspace around the broken solution to the standard charts in the interior of the modulispace) is smooth.

10.3. Gluing constant disks. — The boundary strata of the moduli spaces we studyinvolve splitting off of constant disks and splitting off of disks in the symplectization.In this section we consider gluing constant disks.

We first consider a configuration v as in Theorem 10.3 (a), 10.4 (a), or 10.6 (a1).In all these cases the broken configuration is a two level disk where the second levelconsists of a constant 3-punctured disk v2 that is attached to the first level disk at aLagrangian intersection puncture with asymptotic winding number 1. After we havecarried out the gluing argument in this case we will discuss modifications needed forthe other cases of constant disk gluing.

Assume that the first level disk v1 has m Lagrangian intersection punctures. Wetake the domain of v1 to be the standard domain ∆1 ≈ ∆m. (As explained in Sec-tion 9.4, we may assume that the domain is stable by adding extra marked pointsnear the positive puncture.) Recall that we defined a functional analytic neighborhoodW (a;n) of v1, where W (a;n) is a product of an infinite dimensional weighted Sobolevspace H (a; δ,n) and a finite dimensional space which is an open neighborhood B ofthe origin in Rm−2 × R× Rm, see Section 9.2. Here the first Rm−2-component of anelement in B corresponds to variations of the conformal structure of ∆1, the secondR-factor to shifts of the map in the symplectization direction near the positive punc-ture, and the last Rm-factor corresponds to shifts along the knot near the Lagrangianintersection punctures. Here we will write W 1 for this neighborhood W (a;n) andthink of it as a product

W 1 = W 10 ×B1

where B1 is an open subset of R, as follows. Let q denote the negative puncture wherethe second level disk is attached. Then B1 corresponds to shifts along the knot at q.

Consider the negative puncture q at which the constant three punctured diskv2 : ∆2 → K, where ∆2 ≈ ∆3 is a standard domain with three punctures, is attachedand fix a half-strip neighborhood Q = (−∞, 0]×[0, 1] of it such that v1(Q) lies entirelyin the standard neighborhood of K with complex analytic coordinates.

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For ρ > 0, define a standard domain ∆ρ ≈ ∆m+1 as follows. Remove the neighbor-hood (−∞,−ρ)×[0, 1] of q from ∆1 and the neighborhood (ρ,∞)×[0, 1] of the positivepuncture in ∆2, getting domains ∆1

ρ and ∆2ρ. The domain ∆ρ is then obtained by iden-

tifying the boundary segments −ρ × [0, 1] ⊂ ∆1ρ and ρ × [0, 1] ⊂ ∆2

ρ. Then ∆ρ

contains the strip Qρ ≈ [−ρ, ρ]× [0, 1]:

Qρ = ∆ρ − (∆10 ∪∆2

0).

We next define a pre-gluing wρ : ∆ρ → T ∗Q (i.e., an approximate solution close tothe broken disk v) and a neighborhood of it in a suitably weighted space of maps. Westart with the map. Fix complex analytic coordinates C×C2 around p ∈ K on T ∗Q,where p is the point where the constant disk v2 sits. Let φ : ∆ρ → C be a smoothfunction which equals 1 on ∆1

ρ/2, equals 0 on ∆2ρ and is real-valued and holomorphic

on the boundary. (Holomorphic on the boundary just means that the restriction of ∂to the boundary vanishes. For example, if s + it ∈ R × [0, 1] are coordinates on thestrip and φ(s) is an ordinary real valued cut-off function then a corresponding complexvalued cut-off function that is holomorphic on the boundary is φ(s) + iψ(t)dφ/ds(s),where ψ(t) is a small function with support near ∂[0, 1] such that ψ(0) = ψ(1) = 0

and dψ/dt(0) = dψ/dt(1) = 1.)Define

wρ(z) =

v1(z), z ∈ ∆1

ρ/2,

φ(z)v1(z) z /∈ ∆1ρ/2,

where the last expression refers to the analytic coordinates around q correspond-ing to 0 in the coordinate system. Note then that wρ takes the boundary ∂∆ρ toL = LK ∪Q and that ∂Jwρ is supported in Qρ. Furthermore, using the Fourier ex-pansion of v1 near q,

v1(z) =∑k>1

cke−kπz,

we find that|∂Jwρ|C1 = O(e−πρ).

Define a weight function λρ : ∆ρ → R as follows, where ηδ : ∆m+1 → R denotes theweight function on ∆1,

λρ(z) =

ηδ(z) for z ∈ ∆1

0,

eδ(ρ−|τ |) for z ∈ Qρ ≈ [−ρ, ρ]× [0, 1],

1 for z ∈ ∆20.

Let ‖ ·‖k,ρ denote the Sobolev norm with k derivatives on ∆ρ and weight function λρ.From the above we then find

(10.3) ‖∂Jwρ‖1,ρ 6 |∂Jwρ|C1

∫ ρ

−ρeδ(ρ−|τ |)dτ = O(e−(π−δ)ρ).

We next define configuration spaces of maps giving neighborhoods of the approxi-mate solutions wρ. As in Section 9.2 this space is a direct sum of an infinite dimensional

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space and two finite dimensional summands. We first discuss the infinite dimensionalsummand.

Define H2,ρ(wρ) as the Sobolev space of vector fields v along wρ (i.e., sections ofw∗ρT (T ∗Q)→ ∆ρ) which satisfies the following requirements.

– If ζ ∈ ∂∆ρ maps to LK (maps to Q) then v(ζ) is tangent to LK (resp. to Q).– ∇v + J ∇v i = 0 along ∂∆ρ.– Fix an endpoint ζ0 ∈ ∂∆ρ of the vertical segment which separates the part of ∆ρ

which corresponds to ∆1 from that corresponding to ∆2. We require that v(ζ0) = 0.

Here the first two requirements have counterparts in Section 8.4 and the third isconnected to the addition of certain cut-off solutions in the gluing region. We endowH2,ρ(wρ) with the weighted Sobolev 2-norm ‖ · ‖2,ρ.

Second, we discuss the finite dimensional factor Bρ = B1ρ × B2

ρ . Here B1ρ is an

open neighborhood of the origin in Rm−2 × R × Rm−1 and agrees with the finitedimensional factor of W 1

0 in the following sense. The first Rm−2-factor corresponds tothe conformal variations of ∆ρ inherited from ∆1, the second R-factor corresponds toshifts at the positive puncture, and the last Rm−1-factor corresponds to shifts alongthe knot K at Lagrangian intersection punctures that are also punctures of ∆1. Thesecond factor B2

ρ is an open neighborhood of the origin in R3×R2×R, where the firstR3-factor corresponds to constant vector fields supported in Qρ along the Lagrangianin a neighborhood of p that are cut off in finite regions near the ends of Qρ, where theweight function λρ is uniformly bounded and where the second factor corresponds tothe shifts along K supported at the Lagrangian intersection punctures that are alsopunctures of ∆2. Finally, the third R-factor is a newborn conformal variation definedas follows.

Consider the domain of the constant disks as a strip R×[0, 1] with positive punctureat +∞, one negative puncture at 0, and one at −∞. Let v be the constant vectorfield ∂τ and note that its flow moves the puncture at 0 and that in the standard modelof the 3-punctured disk this vector field looks like c1+O(e−π|τ |) at the puncture at +∞and at one of the punctures at −∞, whereas it looks like c2e−πτ + O(1) at the otherpuncture at −∞, where cj , j = 1, 2 are real constants. We extend this vector field vholomorphically over the gluing region Qρ and then cut it off using a cut-off function βwith derivative supported near the end of Qρ that comes from ∆1 where the weightfunction λρ is close to 1. The conformal variation is then the complex anti-linear mapi∂(βv).

Note that the conformal variation in ∆ρ that is inherited from the conformal vari-ation at q in ∆1 can be identified with the linear combination of conformal variationsas above for the two punctures from ∆2 which looks like 0 + O(e−π|τ |) at the posi-tive puncture. We take the R-factor to correspond to this variation. (Note that thisconformal variation is supported in ∆1

ρ and agrees with the conformal variation in ∆1

corresponding to the negative puncture q where the constant disk is attached.)

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Remark 10.11. — We note that there is a complementary linear combination of thetwo newborn conformal variations with non-zero leading constant term at the positivepuncture of the constant disk that corresponds to the gluing parameter ρ which, fromthe point of view of the domain, shifts the boundary maximum between the two newpunctures, see Remark 9.4.

Let E1,ρ denote the space of complex anti-linear maps T∆ρm+2 → w∗ρT (T ∗Q), again

weighted by λρ. The linearization of the ∂J -operator at wρ is then an operator

L∂J : H2,ρ(wρ)×Bρ −→ E1,ρ,

Lemma 10.12. — The operators L∂J admit right inverses which are uniformly boundedas ρ→∞.

Proof. — The argument here is standard. Let k1, . . . , kl be a basis of the kernel Kof the linearized operator on v1. Fix a cut-off function β which equals 1 on the partof ∆ρ corresponding to ∆1 and with first and second derivatives supported in Qρ ofsize O(ρ−1). (Such a cut-off function exists since the length of Qρ equals 2ρ.) We willestablish an estimate

(10.4) ‖v‖2,ρ 6 C‖L∂jv‖1,ρ,

where C > 0 is a constant, for v in the L2-complement of the subspace K spannedby the cut-off solutions βk1, . . . , βkl. The lemma follows from this estimate.

We argue by contradiction: assume that the estimate does not hold. Then there isa sequence vρ in this L2-complement with

(10.5) ‖vρ‖2,ρ = 1, and ‖L∂Jvρ‖1,ρ −→ 0.

We write vρ = uρ + b1ρ + b2ρ, where uρ ∈ H2,ρ(wρ), b1ρ ∈ B1ρ , and b2ρ ∈ B2

ρ . Fixcut-off functions βj on ∆ρ, j = 1, 2 with the following properties. The function β1

equals 1 on ∆1ρ/2 ⊂ ∆ρ and equals 0 on ∆2

ρ ⊂ ∆ρ. Furthermore, β1uρ is holomorphicon the boundary, and |Dβ1| = O(ρ−1). The function β2 has similar properties butwith support in ∆2

ρ ⊂ ∆ρ. We also let α be a similar cut-off function on ∆ρ, equalto 1 on Qρ/2 and equal to 0 outside Qρ−1.

Since |∂βρj | → 0 as ρ→ 0 we then have∥∥∥L∂J(β1uρ + b1ρ + b2ρ|∆1ρ)∥∥∥

1,ρ6 |∂βρj | ‖uρ‖1,ρ +

∥∥∥L∂J(uρ + b1ρ + b2ρ|∆1ρ)∥∥∥

1,ρ

6 |∂βρj | ‖uρ‖1,ρ + ‖L∂Jvρ‖1,ρ −→ 0,

as ρ → ∞. We then conclude from transversality of v1 (i.e., invertibility of the lin-earized operator off of its kernel) that there exists a constant M > 0 such that

(10.6)∥∥∥β1uρ + b1ρ + b2ρ|∆1

ρ

∥∥∥2,ρ6M

∥∥∥L∂J(β1uρ + b1ρ + b2ρ|∆1ρ)∥∥∥

1,ρ−→ 0.

In particular the cut-off constant solution in the gluing region goes to 0.Similarly we have ∥∥∥L∂J(β2uρ + b2ρ|∆2

ρ)∥∥∥

1,ρ−→ 0.

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We conclude from the invertibility of the standard operator on the three punctureddisk that

(10.7)∥∥∥β2uρ + b2ρ|∆2

ρ

∥∥∥2,ρ−→ 0.

After dividing the weight function in the gluing region Qρ/2 ≈ [−ρ2 ,ρ2 ] × [0, 1]

by its maximum the problem on the gluing region converges to the ∂-problem onthe strip with R3-boundary condition and negative exponential weights at both ends(i.e., weight function ρ(s+ it) = e−δ|s|). This problem has a three-dimensional kernelspanned by constant solutions in R3. As mentioned above, the estimates (10.6) and(10.7) imply that the components along the constant solutions go to zero. This givesfirst that ∥∥L∂J(αuρ)

∥∥1,ρ−→ 0,

and then, by invertibility of L∂J on the complement of the kernel, also that‖αuρ‖2,ρ → 0. Our assumption thus implies that ‖vρ‖2,ρ → 0. This contradicts(10.5). The lemma follows.

The next thing to establish is the quadratic estimate for the non-linear term in theTaylor expansion of ∂J around wρ, i.e., around the origin in H2,ρ × Bρ. We use theexponential map as in Section 9.2 to define the local coordinate system around wρand the estimate for the non-linear term follows from a standard argument that usesthe uniform bounds on the derivatives of the exponential map in our metric, see[18, Lem.A.18] and also [13, 15]. In fact the standard argument gives the correspond-ing unweighted estimate but then the case of positive weights follows since the lefthand side of the inequality is linear in the weight whereas the right hand side isquadratic. So the inequality follows for weights bounded from below. Note also thatvariations along the cut-off solutions in Bρ give contributions to the non-linear termonly in the regions where the derivatives of the cut-off functions are supported andin such regions the weight functions have finite size.

Remark 10.13. — It is essential here that the cut-off solutions are real solutions tothe non-linear equation since a small error term would give a large norm contributionbecause of the large weight function in Nρ, which in turn is key for the proof of theuniform invertibility of the differential in Lemma 10.12.

The final step is then to show surjectivity of the construction. More concretely,this means that we must show that any sequence of disks which converges in thesense of Section 8.5 to a broken disk eventually lies in a small ‖ · ‖2,ρ-neighborhoodof wρ. This follows once we show that any holomorphic disk in a C0-neighborhood ofthe approximate solution is also close in ‖ · ‖2,ρ-norm. The proof of that fact followsfrom the knowledge of explicit solutions in the region where the weight is big. HereC0-control at the ends gives norm control, see [18, Proof of Th.A.21] or [9, Proof ofTh. 1.3]. This finishes the gluing results needed in the cases when we glue one constant3-punctured disks at a Lagrangian intersection puncture of winding number 1.

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The remaining cases for gluing constant disks are proved by modifications of theabove argument that we describe next. Consider first Theorem 10.6 (a2). Here wereplace the gluing parameter ρ with two independent gluing parameters (ρ1, ρ2) ∈[0,∞)2, one for each constant disk. Likewise we have two copies of the new finitedimensional factors in the configuration space. The gluing argument is then a wordby word repetition of the above.

Next consider broken disks as in Theorem 10.4(c). Here the exponential weightat the winding 3

2 -puncture of v1 is δ ∈ (π/2, π) and the boundary condition in thestrip Qρ has different constant Lagrangians along the two boundary components. Thecut-off solutions in B2

ρ change accordingly: instead of an R3-factor of cut-off solutionswe have an R5-factor, R5 = R × R2 × R2. The R-factor is a constant solution in thedirection of the knot. The first R2-factor contains cut-off solutions near the positivepuncture of ∆2

ρ of the form ceπz/2 for c a vector in the appropriate Lagrangian 2-spaceperpendicular to the knot, the second R2-factor consists of cut-off solutions of theform ce−πz/2. Then in Lemma 10.12 we replace (10.7) with the estimate on the threepunctured disk with boundary condition corresponding to the constant disk. I.e., indirections perpendicular to the knot the boundary condition are two perpendicularLagrangian planes at the two boundary components near the positive puncture andone of these planes between the two negative punctures. There is a small positiveexponential weight at the negative punctures, the weight δ and two cut-off solutionsat the positive puncture. In the directions perpendicular to the knot the ∂-operatoris then an isomorphism and the argument above proceeds as before.

Remark 10.14. — In Theorem 10.3 (c) there are two different constant disks and thecorresponding boundary points cancel out. Geometrically this corresponds to pushinga winding 1

2 puncture through a winding 1 puncture.

Finally, we consider Theorem 10.6 (d2). The argument here is the same as that justdescribed for Theorem 10.4 (c) with the only difference being that the 3-puncturedconstant disk should be replaced by a 4-punctured disk and that we invert the operatoron the L2 complement of the additional conformal variation in the 4-punctured disk.In fact, when the 4-punctured disk is broken into two levels it corresponds to the3-level configuration with the two top levels as in Theorem 10.4 (c) and a third levelconstant disk attached at the winding 1 puncture of the second level constant disk.

10.4. Symplectization gluing. — Consider a disk with two non-constant levels as inTheorem 10.6 (b) or (c), Theorem 10.3 (b) or (c), or Theorem 10.4 (b). The argumentneeded to glue such configurations is similar to the one in Section 10.3 and we onlysketch the details. There are again four steps: define an approximate solution, proveuniform invertibility of the differential, establish a quadratic estimate for the non-linear term, and show surjectivity of the construction.

We consider first the case when we glue a symplectization disk to a disk in T ∗Qand discuss modifications needed when the second level also lives in the symplectiza-tion later. Denote the top-level disk in the symplectization v1 : ∆1 = ∆m → R×S∗Q

J.É.P. — M., 2017, tome 4


and the m second level disks v2,j : ∆mj → T ∗Q, j = 1, . . . ,m. Recall that by addingmarked points we reduce to the case when all domains involved are stable, see Sec-tion 9.4.

Each symplectization disk lies in a natural R-family. Let t denote a standard co-ordinate on the R-factor. Fix the unique map v1 in this family that takes the largestboundary maximum in ∆1 to the slice t = 0. By asymptotics at the negative punc-tures, for all T > 0 sufficiently large (v1)−1(t 6 −T) consists of m half strip regionswith one component around each negative puncture of v1. Furthermore, as T → ∞the inverse image of the slice t = −T converges to vertical segments at an expo-nential rate (since the map agrees with trivial Reeb chord strips up to exponentialerror). We fix such a slice and consider the vertical segments through its end point.Parameterize the neighborhoods of all the punctures cut at these vertical segmentsby (−∞, 0] × [0, 1]. For ρ > 0, let ∆1

ρ ⊂ ∆1 be the subset obtained by removing(−∞,−ρ)× [0, 1] from the neighborhood (−∞, 0]× [0, 1] of each negative puncture.

Fix neighborhoods [0,∞)× [0, 1] of the positive puncture in each ∆2,j , j = 1, . . . ,m

in which the map is well approximated by the trivial strip at the positive punctureand let ∆2,j

ρ ⊂ ∆2,j denote the subset obtained by removing (ρ,∞) × [0, 1] fromthis neighborhood. Let ∆ρ denote the domain obtained by adjoining ∆2,j

ρ to ∆1ρ by

identifying the vertical segment at the positive puncture of ∆2,jρ with the vertical

segment of the negative puncture in ∆1ρ where v2,j is attached to v1. Then we get m

strip regions Qjρ = [−ρ, ρ]× [0, 1] ⊂ ∆ρ around each vertical segment where the diskswere joined.

By interpolating between the two maps joined at each negative puncture using thestandard coordinates near the Reeb chords we find a pregluing

wρ : ∆ρ −→ T ∗Q

such that ∂Jwρ is supported only in the middle [−1, 1] × [0, 1] of each Qjρ and suchthat

|∂Jwρ|C1 = O(e−αρ),

where α > 0 depends on the angle between the Lagrangian subspaces of the contacthyperplane obtained by moving the tangent space of ΛK at the Reeb chord start pointto the tangent space of ΛK at the Reeb chord end point by the linearized Reeb flow.

As in Section 10.3 we use a configuration space of maps in a neighborhood of wρthat is a product of an infinite and a finite dimensional space of functions. We firstconsider the infinite dimensional factor. Define weight functions λρ : ∆ρ → R bypatching (suitably scaled) weight functions ηδ of the domains of the broken diskswhere we take 0 < δ < α. In particular, we have λρ(τ + it) = cje

δ|τ | for τ + it ∈ Qjρ.Then, writing ‖ · ‖k,ρ for the Sobolev k-norm with this weight, we have

‖∂Jwρ‖1,ρ = O(e(δ−α)ρ).

We let H2,ρ(wρ) denote the λρ-weighted Sobolev space of vector fields along wρwhich are tangent to the Lagrangians, holomorphic on the boundary, and which sat-isfy the following vanishing condition. The map wρ maps the strip regions Qjρ into

J.É.P. — M., 2017, tome 4


small neighborhoods of the Reeb chord strips where we have standard coordinatesR× (−ε, L+ ε)× C2 and we require that the R-component of the vector field van-ishes at one of the endpoints of the vertical segments where the disks were joined.Thus there are in total m vanishing conditions.

Next we discuss the finite dimensional factor Bρ = B0ρ × B1

ρ × B2ρ . The second

factor B1ρ is an open subset of the origin in R corresponding to the shift at the

positive puncture of wρ. The third factor B2ρ contains all the conformal variations and

the shifts inherited from the negative punctures of the second level disks. Thus B2ρ is

a neighborhood of the origin in

Πmj=1(Rmj−2 × Rmj ).

Finally, the first factor B0ρ is an open subset of the origin in a codimension one

subspace of(R× R2)m,

where each (R × R2)-factor corresponds to a specific second level disk. TheR-component of the jth puncture of v1 corresponds to a cut-off shifting vectorfield aj in the R-direction of the symplectization supported in Qjρ. The R2-componentcorresponds to the two newborn conformal variations in ∆2,j

ρ . As before these con-formal variations have the form γ = ∂V where V is a vector field along ∆ρ. The firstfactor of R2 corresponds to a variation γj1 that agrees with the conformal variation atthe negative puncture in ∆1 where v2,j is attached. The second factor is spanned byγ2,j = ∂V2 where V2 is the vector field in ∆2,j

ρ ∪Qjρ that corresponds to translationsalong the real axis that moves all the boundary maxima in ∆2,j

ρ cut off near the endof Qρ in ∆1

ρ. The codimension one subspace is the orthogonal complement of the linegiven by the equation

γ2,1 = γ2,2 = · · · = γ2,m.

Note that this later conformal variation corresponds to changing ρ.

Remark 10.15. — The nature of the conformal variations γ1,j and γ2,j are easy tosee using a different conformal model for the domain ∆ρ as follows. Consider thedomain of ∆1 as the upper half plane H with positive puncture at ∞ and negativepunctures along the real axis. The conformal variations of this domain can be viewedas translating the negative punctures along the real axis. To construct the domain ∆ρ

we think also of the domains ∆2,j as upper half planes. Cut out small half disksof radius cje−αρ near the negative punctures of ∆1 and glue in the half disks inthe domain ∆2,j of radius cjeαρ scaled by e−2αρ. Now the conformal variation γ1,j

corresponds to translating the whole half disk at the jth negative puncture of ∆1

rigidly in the real direction and the conformal variation γ2,j corresponds to keepingthe small half disk fixed but scaling it so that its negative punctures move closertogether.

We use the neighborhood Wρ = H2,ρ × Bρ of wρ. In order to apply Lemma 10.10we must first establish the counterpart of Lemma 10.12. Here we invert the linearized

J.É.P. — M., 2017, tome 4


operator on the L2-complement of the subspace spanned by cut-off kernel elementsin ∆1 and ∆2,j defined as follows. The infinite dimensional components are indeed justa cut-off vector field. For the finite dimensional components we identify the conformalvariation at the jth negative puncture of ∆1 with γ1,j , the shift at this negativepuncture with aj , and the shift at the positive puncture of ∆2,j with γ2,j . To showuniform invertibility we then argue by contradiction as in the proof of Lemma 10.12.Using the above identifications of finite dimensional factors, the result follows in astraightforward way.

Finally, the two remaining steps, the quadratic estimate for the non-linear termand the surjectivity of the construction are completely analogous to their counterpartsin Section 10.3 and will not be discussed further.

In the case that the second level disk lies in the symplectization as well we startas above by fixing a representative for v1 and a slice t = −T after which thisrepresentative is well approximated by Reeb chord strips. We then fix representativesfor all the non-trivial second level curves v2,j (of which there is only one in our case)that are translated sufficiently much so that they are well approximated by Reebchord strips in the slice t = −T at their positive punctures. We then repeat theargument above.

10.5. Point constraints on the knot. — An analogous construction allows us toexpress neighborhoods of disks with Lagrangian intersection punctures of windingnumber 1 inside the space of disks with these punctures removed. In the analyticalC× C2-coordinates around the knot a disk v with such a puncture looks like

v(z) =∑n>0

cne−nπz, z ∈ [0,∞)× [0, 1], cn ∈ R3 or cn ∈ R× iR2

with c0 = (c′0, 0), whereas a general disk looks the same way but has unrestricted c0.We can thus construct a configuration space W for unrestricted disks in a neighbor-hood of v as

W = W ′ ⊕ R2,

where W ′ is the configuration space for disks in a neighborhood of v with Lagrangianintersection puncture of winding number 1 and R2 is spanned by two cut-off constantsolutions in the Lagrangian perpendicular toK. The zero-set of the ∂J -operator actingon W then gives a neighborhood of v in the space of unrestricted disks.

10.6. Proofs of the structure theorems. — The proof of all the theorems on thestructure of the compactified moduli spaces as manifolds with boundary with cornersnow follow the same pattern. Transversality and compactness results give the possibledegenerations and gluing give neighborhoods of several level disks in the boundary.The manifold structure in the interior is a consequence of standard Fredholm theory,whereas charts near the boundary are obtained from the conformal structures of thedomains.

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Proof of Theorem 10.1. — Part (i) follows immediately from Lemma 9.5 and Theorem8.12. Consider part (ii). Lemma 9.5 and Theorem 8.12 imply that the broken diskslisted are the only possible configurations in the boundary of the compactified modulispace. It follows from (the parameterized version of) Lemma 10.10 that the gluingparameter gives a parameterization of the boundary of the reduced moduli space.Recall that we identified the gluing parameter with a certain conformal variation(that shifts all the boundary maxima in the second level disk) and we topologize aneighborhood of the broken configuration using the induced map to the compactifiedspace of conformal structures. This establishes (ii).

Proof of Theorem 10.2. — The theorem follows immediately from Lemma 9.5 andTheorem 8.12.

Proof of Theorem 10.3. — The proof is analogous to the proof of Theorem 10.1 (ii)except for (c). Here a disk without Lagrangian intersection punctures moves out asa rigid disk in the symplectization into the R-invariant region and the translationsalong R give a neighborhood of the boundary.

Proof of Theorem 10.4. — The argument is analogous to the proofs above and weexplain only how to parameterize the boundary in the cases that differ from theabove. Consider (b). Recall that we identified the gluing parameter with the con-formal variation that translates all the boundary maxima in the second level disksuniformly. As above we use this to parameterize a neighborhood of the boundary.Finally, consider (c). Here again the boundary can be parameterized by the gluingparameter which corresponds to a conformal variation. In particular, the boundarypoint corresponds to a three punctured disk splitting off. As explained in Remark10.14 there are two such disks and the corresponding boundaries of the moduli spacenaturally fit together to a smooth 1-manifold.

Remark 10.16 (cf. Remark 10.5). — Consider a holomorphic disk near the codimen-sion one boundary as in Theorem 10.4 (c). Remark 8.13 gives a local model (4.1) forthe disk, parameterized by a half disk in the upper half plane near the two collidingcorners with one puncture at 0 and one at ε > 0. The above proof shows that thenewborn conformal variation which here is the length of the stretching strip can beused as local coordinate in the moduli space near the corner. A conformal map thattakes a vertical segment in the stretching strip to the upper arc in the unit circle andthe boundary of the domain in the disk splitting off to the real line gives a smoothchange of coordinates from this parameter to the coordinates given by ε. Thus thelocal model (4.1) used in the definition of the string operations is Ck-close to theactual moduli space, when both are viewed as parameterized by the coordinates ε. Asimilar discussion applies to Theorem 10.4 (c), using the local model (4.2) with δ = 0.

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Proof of Theorem 10.6. — Arguments for producing neighborhoods of codimensionone boundary strata are similar to the above, so we discuss the codimension two parts.

Consider a broken disk as in (a2). The gluing result needed in this case is analogousto the argument in Section 10.3. Here however we attach two constant disks, producingapproximate solutions wρ1,ρ2 depending on two independent variables ρ1, ρ2 →∞. Inthis case there are two independent newborn conformal variations and the linearized∂J -operator is inverted on the complement of their linear span. It follows as abovethat the projection of the moduli space is an embedding into the space of conformalstructures and we induce the corner structure from there. Note that this is coherentwith our treatment of nearby codimension one boundary disks.

The arguments in cases (b2), (c2), and (d2) follow the same lines. We produceapproximate solutions depending on two independent variables. In case (b2) the lin-earized operator is inverted on the complement of the 2-dimensional spaces spannedby the cut off shift of the symplectization disk and the newborn conformal structureof the constant disk. In case (c2) the linearized operator is inverted on the comple-ment of the (independent) shifts of the first and second level disks, and in case (d2)on the complement of the newborn conformal structure and the additional conformalstructure in the constant 4-punctured disk. In all cases, the corner structure is in-duced from the corresponding structure on the space of conformal structures and theconstruction is compatible with nearby strata of lower codimension.

Remark 10.17 (cf. Remark 10.7). — Consider a holomorphic disk near the codimen-sion two corner as in Theorem 10.6 (d2). Remark 8.13 gives a local model (4.2) forthe disk, parameterized by a half disk in the upper half plane near the three collidingcorners with one puncture at 0 and the two others at boundary points δ < 0 andε > 0. The above proof shows that the newborn conformal variation (which here isthe length of the stretching strip) together with the difference between the boundarymaxima in the 4-punctured disk splitting off can be used as local coordinates in themoduli space near the corner. A conformal map that takes a vertical segment in thestretching strip to the upper arc in the unit circle and the boundary of the domain inthe disk splitting off to the real line gives a smooth change of coordinates from thesetwo parameters to the coordinates given by (ε, δ). Thus the local model (4.2) used inthe definition of the string operations is Ck-close to the actual moduli space, whenboth are viewed as parameterized by the coordinates (ε, δ).

Proof of Theorem 10.8. — The theorem follows from the discussion in Section 10.5.

Proof of Theorem 10.9. — The theorem follows from the discussion in Section 10.5 incombination with the argument in the proof of Theorem 10.4 (c).

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Manuscript received February 5, 2016accepted May 23, 2017

Kai Cieliebak, Institut für Mathematik, Universität Augsburg86135 Augsburg, GermanyE-mail : [email protected] : https://www.math.uni-augsburg.de/prof/geo/mitarbeiter/cieliebak/

Tobias Ekholm, Department of Mathematics, Uppsala University751 06 Uppsala, SwedenandInstitut Mittag-LefflerAurav 17, 182 60 Djursholm, SwedenE-mail : [email protected]

Janko Latschev, Universität Hamburg, Fachbereich MathematikBundesstraße 55, 20146 Hamburg, GermanyE-mail : [email protected] : https://www.math.uni-hamburg.de/home/latschev/

Lenhard Ng, Department of Mathematics, Duke UniversityDurham, NC 27708-0320, USAE-mail : [email protected] : http://alum.mit.edu/www/ng

J.É.P. — M., 2017, tome 4


mailto:[email protected]

https://www.math.uni-augsburg.de/prof/geo/mitarbeiter/cieliebak/


mailto: [email protected]

https://www.math.uni-hamburg.de/home/latschev/


http://alum.mit.edu/www/ng

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